thư viện số dau prealgebra

In the following exercises, find the prime factorization of each number using the factor tree method.. 267.[r]

Prealgebra

SENIOR CONTRIBUTING AUTHORS

LYNN MARECEK,SANTA ANA COLLEGE

6100 Main Street MS-375 Houston, Texas 77005

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1 Preface 1

Whole Numbers 7

1.1 Introduction to Whole Numbers

1.2 Add Whole Numbers 24

1.3 Subtract Whole Numbers 40

1.4 Multiply Whole Numbers 55

1.5 Divide Whole Numbers 73

The Language of Algebra 101 2.1 Use the Language of Algebra 101

2.2 Evaluate, Simplify, and Translate Expressions 121

2.3 Solving Equations Using the Subtraction and Addition Properties of Equality 135

2.4 Find Multiples and Factors 149

2.5 Prime Factorization and the Least Common Multiple 163

Integers 183

3.1 Introduction to Integers 183

3.2 Add Integers 201

3.3 Subtract Integers 216

3.4 Multiply and Divide Integers 236

3.5 Solve Equations Using Integers; The Division Property of Equality 250

Fractions 273

4.1 Visualize Fractions 273

4.2 Multiply and Divide Fractions 297

4.3 Multiply and Divide Mixed Numbers and Complex Fractions 317

4.4 Add and Subtract Fractions with Common Denominators 330

4.5 Add and Subtract Fractions with Different Denominators 342

4.6 Add and Subtract Mixed Numbers 364

4.7 Solve Equations with Fractions 380

Decimals 407 5.1 Decimals 407

5.2 Decimal Operations 426

5.3 Decimals and Fractions 445

5.4 Solve Equations with Decimals 459

5.5 Averages and Probability 468

5.6 Ratios and Rate 481

5.7 Simplify and Use Square Roots 494

Percents 517

6.1 Understand Percent 517

6.2 Solve General Applications of Percent 534

6.3 Solve Sales Tax, Commission, and Discount Applications 546

6.4 Solve Simple Interest Applications 560

6.5 Solve Proportions and their Applications 570

The Properties of Real Numbers 595 7.1 Rational and Irrational Numbers 595

7.2 Commutative and Associative Properties 604

7.3 Distributive Property 617

7.4 Properties of Identity, Inverses, and Zero 629

9

10

11

8.4 Solve Equations with Fraction or Decimal Coefficients 705

Math Models and Geometry 721 9.1 Use a Problem Solving Strategy 721

9.2 Solve Money Applications 736

9.3 Use Properties of Angles, Triangles, and the Pythagorean Theorem 749

9.4 Use Properties of Rectangles, Triangles, and Trapezoids 768

9.5 Solve Geometry Applications: Circles and Irregular Figures 797

9.6 Solve Geometry Applications: Volume and Surface Area 809

9.7 Solve a Formula for a Specific Variable 830

Polynomials 855

10.1 Add and Subtract Polynomials 855

10.2 Use Multiplication Properties of Exponents 866

10.3 Multiply Polynomials 881

10.4 Divide Monomials 895

10.5 Integer Exponents and Scientific Notation 915

10.6 Introduction to Factoring Polynomials 933

Graphs 955

11.1 Use the Rectangular Coordinate System 955

11.2 Graphing Linear Equations 978

11.3 Graphing with Intercepts 996

11.4 Understand Slope of a Line 1013 A Cumulative Review 1051 B Powers and Roots Tables 1057 C Geometric Formulas 1061

PREFACE

Welcome to Prealgebra, an OpenStax resource This textbook was written to increase student access to high-quality learning materials, maintaining highest standards of academic rigor at little to no cost

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Format

You can access this textbook for free in web view or PDF through openstax.org

AboutPrealgebra

Prealgebra is designed to meet scope and sequence requirements for a one-semester prealgebra course The text introduces the fundamental concepts of algebra while addressing the needs of students with diverse backgrounds and learning styles Each topic builds upon previously developed material to demonstrate the cohesiveness and structure of mathematics

Students who are takingBasic MathematicsandPrealgebraclasses in college present a unique set of challenges Many students in these classes have been unsuccessful in their prior math classes They may think they know some math, but their core knowledge is full of holes Furthermore, these students need to learn much more than the course content They need to learn study skills, time management, and how to deal with math anxiety Some students lack basic reading and arithmetic skills The organization ofPrealgebramakes it easy to adapt the book to suit a variety of course syllabi

Coverage and Scope

Prealgebrafollows a nontraditional approach in its presentation of content The beginning, in particular, is presented as a sequence of small steps so that students gain confidence in their ability to succeed in the course The order of topics was carefully planned to emphasize the logical progression throughout the course and to facilitate a thorough understanding of each concept As new ideas are presented, they are explicitly related to previous topics

Chapter 1: Whole Numbers

Each of the four basic operations with whole numbers—addition, subtraction, multiplication, and division—is modeled and explained As each operation is covered, discussions of algebraic notation and operation signs, translation of algebraic expressions into word phrases, and the use the operation in applications are included

Chapter 2: The Language of Algebra

evaluating, and translating algebraic expressions The Division Property of Equality is introduced and used to solve one-step equations

Chapter 4: Fractions

Fraction circles and bars are used to help make fractions real and to develop operations on them Students continue simplifying and evaluating algebraic expressions with fractions, and learn to use the Multiplication Property of Equality to solve equations involving fractions

Chapter 5: Decimals

Basic operations with decimals are presented, as well as methods for converting fractions to decimals and vice versa Averages and probability, unit rates and unit prices, and square roots are included to provide opportunities to use and round decimals

Chapter 6: Percents

Conversions among percents, fractions, and decimals are explored Applications of percent include calculating sales tax, commission, and simple interest Proportions and solving percent equations as proportions are addressed as well

Chapter 7: The Properties of Real Numbers

The properties of real numbers are introduced and applied as a culmination of the work done thus far, and to prepare students for the upcoming chapters on equations, polynomials, and graphing

Chapter 8: Solving Linear Equations

A gradual build-up to solving multi-step equations is presented Problems involve solving equations with constants on both sides, variables on both sides, variables and constants on both sides, and fraction and decimal coefficients

Chapter 9: Math Models and Geometry

The chapter begins with opportunities to solve “traditional” number, coin, and mixture problems Geometry sections cover the properties of triangles, rectangles, trapezoids, circles, irregular figures, the Pythagorean Theorem, and volumes and surface areas of solids Distance-rate-time problems and formulas are included as well

Chapter 10: Polynomials

Adding and subtracting polynomials is presented as an extension of prior work on combining like terms Integer exponents are defined and then applied to scientific notation The chapter concludes with a brief introduction to factoring polynomials

Chapter 11: Graphs

This chapter is placed last so that all of the algebra with one variable is completed before working with linear equations in two variables Examples progress from plotting points to graphing lines by making a table of solutions to an equation Properties of vertical and horizontal lines and intercepts are included Graphing linear equations at the end of the course gives students a good opportunity to review evaluating expressions and solving equations

All chapters are broken down into multiple sections, the titles of which can be viewed in the Table of Contents

Accuracy of Content

We have taken great pains to ensure the validity and accuracy of this text Each chapter’s manuscript underwent rounds of review and revision by a panel of active instructors Then, prior to publication, a separate team of experts checked all text, examples, and graphics for mathematical accuracy A third team of experts was responsible for the accuracy of the Answer Key, dutifully re-working every solution to eradicate any lingering errors Finally, the editorial team conducted a multi-round post-production review to ensure the integrity of the content in its final form

Pedagogical Foundation and Features

Learning Objectives

Each chapter is divided into multiple sections (or modules), each of which is organized around a set of learning objectives The learning objectives are listed explicitly at the beginning of each section and are the focal point of every instructional element

Narrative text

Narrative text is used to introduce key concepts, terms, and definitions, to provide real-world context, and to provide transitions between topics and examples An informal voice was used to make the content accessible to students Throughout this book, we rely on a few basic conventions to highlight the most important ideas:

Key terms are boldfaced, typically when first introduced and/or when formally defined Key concepts and definitions are called out in a blue box for easy reference

approaches that students must master Typically, we include multiple Examples for each learning objective in order to model different approaches to the same type of problem, or to introduce similar problems of increasing complexity All Examples follow a simple two- or three-part format First, we pose a problem or question Next, we demonstrate the Solution, spelling out the steps along the way Finally (for select Examples), we show students how to check the solution Most examples are written in a two-column format, with explanation on the left and math on the right to mimic the way that instructors “talk through” examples as they write on the board in class

Figures

Prealgebracontains many figures and illustrations Art throughout the text adheres to a clear, understated style, drawing the eye to the most important information in each figure while minimizing visual distractions

Supporting Features

Four small but important features serve to support Examples:

Be Prepared!

Each section, beginning with Section 1.2, starts with a few “Be Prepared!” exercises so that students can determine if they have mastered the prerequisite skills for the section Reference is made to specific Examples from previous sections so students who need further review can easily find explanations Answers to these exercises can be found in the supplemental resources that accompany this title

How To

A “How To” is a list of steps necessary to solve a certain type of problem A "How To" typically precedes an Example

Try It

A “Try It” exercise immediately follows an Example, providing the student with an immediate opportunity to solve a similar problem In the web view version of the text, students can click an Answer link directly below the question to check their understanding In the PDF, answers to the Try It exercises are located in the Answer Key

Media

The “Media” icon appears at the conclusion of each section, just prior to the Section Exercises This icon marks a list of links to online video tutorials that reinforce the concepts and skills introduced in the section

Disclaimer: While we have selected tutorials that closely align to our learning objectives, we did not produce these tutorials, nor were they specifically produced or tailored to accompanyPrealgebra

Section Exercises

Each section of every chapter concludes with a well-rounded set of exercises that can be assigned as homework or used selectively for guided practice Exercise sets are named Practice Makes Perfect to encourage completion of homework assignments

Exercises correlate to the learning objectives This facilitates assignment of personalized study plans based on individual student needs

Exercises are carefully sequenced to promote building of skills

Values for constants and coefficients were chosen to practice and reinforce arithmetic facts Even and odd-numbered exercises are paired

Exercises parallel and extend the text examples and use the same instructions as the examples to help students easily recognize the connection

Applications are drawn from many everyday experiences, as well as those traditionally found in college math texts

Chapter Review Features

The end of each chapter includes a review of the most important takeaways, as well as additional practice problems that students can use to prepare for exams

Key Termsprovides a formal definition for each bold-faced term in the chapter

Key Concepts summarizes the most important ideas introduced in each section, linking back to the relevant Example(s) in case students need to review

Chapter Review Exercisesincludes practice problems that recall the most important concepts from each section

Practice Testincludes additional problems assessing the most important learning objectives from the chapter

Answer Keyincludes the answers to all Try It exercises and every other exercise from the Section Exercises, Chapter Review Exercises, and Practice Test

Additional Resources

Student and Instructor Resources

We’ve compiled additional resources for both students and instructors, including Getting Started Guides, manipulative mathematics worksheets, Links to Literacy assignments, and an answer key to Be Prepared Exercises Instructor resources require a verified instructor account, which can be requested on your openstax.org log-in Take advantage of these resources to supplement your OpenStax book

Partner Resources

OpenStax Partners are our allies in the mission to make high-quality learning materials affordable and accessible to students and instructors everywhere Their tools integrate seamlessly with our OpenStax titles at a low cost To access the partner resources for your text, visit your book page on openstax.org

About the Authors

Senior Contributing Authors

Lynn Marecek and MaryAnne Anthony-Smith have been teaching mathematics at Santa Ana College for many years and have worked together on several projects aimed at improving student learning in developmental math courses They are the authors ofStrategies for Success: Study Skills for the College Math Student,published by Pearson HigherEd

Lynn Marecek, Santa Ana College

Lynn Marecek has focused her career on meeting the needs of developmental math students At Santa Ana College she has been awarded the Distinguished Faculty Award, Innovation Award, and the Curriculum Development Award four times She is a Coordinator of Freshman Experience Program, the Department Facilitator for Redesign, and a member of the Student Success and Equity Committee and the Basic Skills Initiative Task Force Lynn holds a bachelor’s degree from Valparaiso University and master’s degrees from Purdue University and National University

MaryAnne Anthony-Smith, Santa Ana College

MaryAnne has served as department chair, acting dean, chair of the professional development committee, institutional researcher, and faculty coordinator on several state and federally-funded grants and was a member of AMATYC’s Placement and Assessment Committee She is the community college coordinator of California’s Mathematics Diagnostic Testing Project

Reviewers

Tony Ayers, Collin College Preston Ridge Campus David Behrman, Somerset Community College Brandie Biddy, Cecil College

Bryan Blount, Kentucky Wesleyan College

Steven Boettcher, Estrella Mountain Community College Kimberlyn Brooks, Cuyahoga Community College Pamela Burleson, Lone Star College University Park Tamara Carter, Texas A&M University

Phil Clark, Scottsdale Community College Christina Cornejo, Erie Community College Denise Cutler, Bay de Noc Community College Richard Darnell, Eastern Wyoming College Robert Diaz, Fullerton College

Karen Dillon, Thomas Nelson Community College Valeree Falduto, Palm Beach State

Bryan Faulkner, Ferrum College

Dianne Hendrickson, Becker College Linda Hunt, Shawnee State University Betty Ivory, Cuyahoga Community College Joanne Kendall, Lone Star College System Kevin Kennedy, Athens Technical College Stephanie Krehl, Mid-South Community College Allyn Leon, Imperial Valley College

Gerald LePage, Bristol Community College Laurie Lindstrom, Bay de Noc Community College Jonathan Lopez, Niagara University

Yixia Lu, South Suburban College Mikal McDowell, Cedar Valley College Kim McHale, Columbia College of Missouri Allen Miller, Northeast Lakeview College

Michelle Moravec, Baylor University TX/McLennan Community College Jennifer Nohai-Seaman, Housatonic Community College

Rick Norwood, East Tennessee State University Linda Padilla, Joliet Junior College

Kelly Proffitt, Patrick Henry Community College Teresa Richards, Butte-Glenn Community College

Christian Roldan-Johnson, College of Lake County Community College Patricia C Rome, Delgado Community College, City Park Campus Kegan Samuel, Naugatuck Valley Community College

Bruny Santiago, Tarrant College Southeast Campus Sutandra Sarkar, Georgia State University

Richard Sgarlotti, Bay Mills Community College Chuang Shao, Rose State College

Carla VanDeSande, Arizona State University

Shannon Vinson, Wake Technical Community College Maryam Vulis, Norwalk Community College

Figure 1.1 Purchasing pounds of fruit at a fruit market requires a basic understanding of numbers (credit: Dr Karl-Heinz Hochhaus, Wikimedia Commons)

Chapter Outline

1.1Introduction to Whole Numbers

1.2Add Whole Numbers

1.3Subtract Whole Numbers

1.4Multiply Whole Numbers

1.5Divide Whole Numbers Introduction

Even though counting is first taught at a young age, mastering mathematics, which is the study of numbers, requires constant attention If it has been a while since you have studied math, it can be helpful to review basic topics In this chapter, we will focus on numbers used for counting as well as four arithmetic operations—addition, subtraction, multiplication, and division We will also discuss some vocabulary that we will use throughout this book

1.1 Introduction to Whole Numbers Learning Objectives

By the end of this section, you will be able to:

Identify counting numbers and whole numbers Model whole numbers

Identify the place value of a digit Use place value to name whole numbers Use place value to write whole numbers Round whole numbers

Identify Counting Numbers and Whole Numbers

Learning algebra is similar to learning a language You start with a basic vocabulary and then add to it as you go along You need to practice often until the vocabulary becomes easy to you The more you use the vocabulary, the more familiar it becomes

Algebra uses numbers and symbols to represent words and ideas Let’s look at the numbers first The most basic numbers used in algebra are those we use to count objects: 1, 2, 3, 4, 5, … and so on These are called thecounting numbers The notation “…” is called an ellipsis, which is another way to show “and so on”, or that the pattern continues endlessly Counting numbers are also called natural numbers

MANIPULATIVE MATHEMATICS

Doing the Manipulative Mathematics activity Number Line-Part will help you develop a better understanding of the counting numbers and the whole numbers

Counting Numbers

The counting numbers start with 1 and continue

1, 2, 3, 4, 5…

Counting numbers and whole numbers can be visualized on anumber lineas shown inFigure 1.2

Figure 1.2 The numbers on the number line increase from left to right, and decrease from right to left

The point labeled 0 is called theorigin The points are equally spaced to the right of 0 and labeled with the counting numbers When a number is paired with a point, it is called thecoordinateof the point

The discovery of the number zero was a big step in the history of mathematics Including zero with the counting numbers gives a new set of numbers called thewhole numbers

Whole Numbers

The whole numbers are the counting numbers and zero

0, 1, 2, 3, 4, 5…

We stopped at 5 when listing the first few counting numbers and whole numbers We could have written more numbers if they were needed to make the patterns clear

EXAMPLE 1.1

Which of the following areⓐcounting numbers?ⓑwhole numbers? 0, 14, 3, 5.2, 15, 105

Solution

ⓐThe counting numbers start at 1, so 0 is not a counting number The numbers 3, 15, and 105 are all counting numbers

ⓑWhole numbers are counting numbers and 0. The numbers 0, 3, 15, and 105 are whole numbers

The numbers 14 and 5.2 are neither counting numbers nor whole numbers We will discuss these numbers later

TRY IT : :1.1 Which of the following areⓐcounting numbersⓑwhole numbers?

TRY IT : :1.2 Which of the following areⓐcounting numbersⓑwhole numbers?

0, 53, 7, 8.8, 13, 201

Model Whole Numbers

Our number system is called aplace value systembecause the value of a digit depends on its position, or place, in a number The number 537 has a different value than the number 735. Even though they use the same digits, their value is different because of the different placement of the 3 and the 7 and the 5.

Money gives us a familiar model of place value Suppose a wallet contains three $100 bills, seven $10 bills, and four $1 bills The amounts are summarized inFigure 1.3 How much money is in the wallet?

Figure 1.3

Find the total value of each kind of bill, and then add to find the total The wallet contains $374.

Base-10 blocks provide another way to model place value, as shown inFigure 1.4 The blocks can be used to represent hundreds, tens, and ones Notice that the tens rod is made up of 10 ones, and the hundreds square is made of 10 tens, or 100 ones

Figure 1.4

Figure 1.5 We use place value notation to show the value of the number 138.

Digit Place value Number Value Total value

1 hundreds 1 100 100

3 tens 3 10 30

8 ones 8 1 +8

Sum =138 EXAMPLE 1.2

Use place value notation to find the value of the number modeled by the base-10blocks shown

Solution

There are 2 hundreds squares, which is 200. There is 1 tens rod, which is 10.

There are 5 ones blocks, which is 5.

Digit Place value Number Value Total value

2 hundreds 2 100 200

1 tens 1 10 10

5 ones 5 1 +5

The base-10 blocks model the number 215. TRY IT : :1.3

Use place value notation to find the value of the number modeled by the base-10 blocks shown

TRY IT : :1.4

Use place value notation to find the value of the number modeled by the base-10 blocks shown

MANIPULATIVE MATHEMATICS

Doing the Manipulative Mathematics activity “Model Whole Numbers” will help you develop a better understanding of place value of whole numbers

Identify the Place Value of a Digit

By looking at money and base-10 blocks, we saw that each place in a number has a different value A place value chart is a useful way to summarize this information The place values are separated into groups of three, called periods The periods areones, thousands, millions, billions, trillions, and so on In a written number, commas separate the periods Just as with the base-10 blocks, where the value of the tens rod is ten times the value of the ones block and the value of the hundreds square is ten times the tens rod, the value of each place in the place-value chart is ten times the value of the place to the right of it

Figure 1.6shows how the number 5,278,194 is written in a place value chart

Figure 1.6

• The digit 7 is in the ten thousands place Its value is 70,000. • The digit 8 is in the thousands place Its value is 8,000. • The digit 1 is in the hundreds place Its value is 100. • The digit 9 is in the tens place Its value is 90. • The digit 4 is in the ones place Its value is 4. EXAMPLE 1.3

In the number 63,407,218; find the place value of each of the following digits:

ⓐ 7 ⓑ 0 ⓒ 1 ⓓ 6 ⓔ 3

Solution

Write the number in a place value chart, starting at the right

ⓐThe 7 is in the thousands place

ⓑThe 0 is in the ten thousands place

ⓒThe 1 is in the tens place

ⓓThe 6 is in the ten millions place

ⓔThe 3 is in the millions place

TRY IT : :1.5 For each number, find the place value of digits listed: 27,493,615

ⓐ 2 ⓑ 1 ⓒ 4 ⓓ 7 ⓔ 5

TRY IT : :1.6 For each number, find the place value of digits listed: 519,711,641,328

ⓐ 9 ⓑ 4 ⓒ2 ⓓ6 ⓔ7

Use Place Value to Name Whole Numbers

So the number 37,519,248 is written thirty-seven million, five hundred nineteen thousand, two hundred forty-eight Notice that the wordandis not used when naming a whole number

EXAMPLE 1.4

Name the number 8,165,432,098,710 in words

Solution

Begin with the leftmost digit, which is It is in the trillions place eight trillion

The next period to the right is billions one hundred sixty-five billion The next period to the right is millions four hundred thirty-two million The next period to the right is thousands ninety-eight thousand

The rightmost period shows the ones seven hundred ten

Putting all of the words together, we write 8,165,432,098,710 as eight trillion, one hundred sixty-five billion, four hundred thirty-two million, ninety-eight thousand, seven hundred ten

TRY IT : :1.7 Name each number in words: 9,258,137,904,061 TRY IT : :1.8 Name each number in words: 17,864,325,619,004 EXAMPLE 1.5

A student conducted research and found that the number of mobile phone users in the United States during one month in 2014 was 327,577,529. Name that number in words

HOW TO : :NAME A WHOLE NUMBER IN WORDS

Starting at the digit on the left, name the number in each period, followed by the period name Do not include the period name for the ones

Use commas in the number to separate the periods Step

Identify the periods associated with the number

Name the number in each period, followed by the period name Put the commas in to separate the periods Millions period: three hundred twenty-seven million

Thousands period: five hundred seventy-seven thousand Ones period: five hundred twenty-nine

So the number of mobile phone users in the Unites States during the month of April was three hundred twenty-seven million, five hundred seventy-seven thousand, five hundred twenty-nine

TRY IT : :1.9 The population in a country is 316,128,839. Name that number.

TRY IT : :1.10 One year is 31,536,000 seconds Name that number.

Use Place Value to Write Whole Numbers

We will now reverse the process and write a number given in words as digits

EXAMPLE 1.6

Write the following numbers using digits

ⓐfifty-three million, four hundred one thousand, seven hundred forty-two

ⓑnine billion, two hundred forty-six million, seventy-three thousand, one hundred eighty-nine Solution

ⓐIdentify the words that indicate periods

Except for the first period, all other periods must have three places Draw three blanks to indicate the number of places needed in each period Separate the periods by commas

Then write the digits in each period

Put the numbers together, including the commas The number is 53,401,742.

ⓑIdentify the words that indicate periods

HOW TO : :USE PLACE VALUE TO WRITE A WHOLE NUMBER

Identify the words that indicate periods (Remember the ones period is never named.) Draw three blanks to indicate the number of places needed in each period Separate the periods by commas

Name the number in each period and place the digits in the correct place value position Step

needed in each period Separate the periods by commas Then write the digits in each period

The number is 9,246,073,189.

Notice that in partⓑ, a zero was needed as a place-holder in the hundred thousands place Be sure to write zeros as needed to make sure that each period, except possibly the first, has three places

TRY IT : :1.11 Write each number in standard form:

fifty-three million, eight hundred nine thousand, fifty-one TRY IT : :1.12 Write each number in standard form:

two billion, twenty-two million, seven hundred fourteen thousand, four hundred sixty-six EXAMPLE 1.7

A state budget was about $77 billion Write the budget in standard form

Solution

Identify the periods In this case, only two digits are given and they are in the billions period To write the entire number, write zeros for all of the other periods

So the budget was about $77,000,000,000.

TRY IT : :1.13 Write each number in standard form:

The closest distance from Earth to Mars is about34 million miles TRY IT : :1.14 Write each number in standard form:

The total weight of an aircraft carrier is 204 million pounds

Round Whole Numbers

In 2013, the U.S Census Bureau reported the population of the state of New York as 19,651,127 people It might be enough to say that the population is approximately 20 million The wordapproximatelymeans that 20 million is not the exact population, but is close to the exact value

The process of approximating a number is calledrounding Numbers are rounded to a specific place value depending on how much accuracy is needed Saying that the population of New York is approximately 20 million means we rounded to the millions place The place value to which we round to depends on how we need to use the number

Figure 1.7 We can see that76 is closer to 80 than to 70.So 76 rounded to the nearest ten is 80.

Now consider the number 72. Find 72 inFigure 1.8

Figure 1.8 We can see that72 is closer to 70, so 72 rounded to the nearest ten is 70.

How we round 75 to the nearest ten Find 75 inFigure 1.9

Figure 1.9 The number 75 is exactly midway between70 and 80.

So that everyone rounds the same way in cases like this, mathematicians have agreed to round to the higher number, 80. So, 75 rounded to the nearest ten is 80.

Now that we have looked at this process on the number line, we can introduce a more general procedure To round a number to a specific place, look at the number to the right of that place If the number is less than 5, round down If it is greater than or equal to 5, round up

So, for example, to round 76 to the nearest ten, we look at the digit in the ones place

The digit in the ones place is a 6. Because 6 is greater than or equal to 5, we increase the digit in the tens place by one So the 7 in the tens place becomes an 8. Now, replace any digits to the right of the 8with zeros So, 76 rounds to

80.

Let’s look again at rounding 72 to the nearest 10. Again, we look to the ones place

EXAMPLE 1.8

Round 843 to the nearest ten

Solution

Locate the tens place

Underline the digit to the right of the tens place Since is less than 5, not change the digit in the tens place

Replace all digits to the right of the tens place with zeros

Rounding 843 to the nearest ten gives 840

TRY IT : :1.15 Round to the nearest ten: 157.

TRY IT : :1.16 Round to the nearest ten: 884. EXAMPLE 1.9

Round each number to the nearest hundred:

ⓐ 23,658 ⓑ 3,978

HOW TO : :ROUND A WHOLE NUMBER TO A SPECIFIC PLACE VALUE

Locate the given place value All digits to the left of that place value not change Underline the digit to the right of the given place value

Determine if this digit is greater than or equal to 5. ◦ Yes—add 1 to the digit in the given place value ◦ No—do not change the digit in the given place value Replace all digits to the right of the given place value with zeros Step

Step Step

ⓐ

Locate the hundreds place

The digit of the right of the hundreds place is Underline the digit to the right of the hundreds place

Since is greater than or equal to 5, round up by adding to the digit in the hundreds place Then replace all digits to the right of the hundreds place with zeros

So 23,658 rounded to the nearest hundred is 23,700

ⓑ

Locate the hundreds place

Underline the digit to the right of the hundreds place

The digit to the right of the hundreds place is Since is greater than or equal to 5, round up by added to the Then place all digits to the right of the hundreds place with zeros

So 3,978 rounded to the nearest hundred is 4,000

TRY IT : :1.17 Round to the nearest hundred: 17,852.

TRY IT : :1.18 Round to the nearest hundred: 4,951. EXAMPLE 1.10

Round each number to the nearest thousand:

Solution

ⓐ

Locate the thousands place Underline the digit to the right of the thousands place

The digit to the right of the thousands place is Since is less than 5, we not change the digit in the thousands place

We then replace all digits to the right of the thousands pace with zeros So 147,032 rounded to the nearest thousand is 147,000

ⓑ

Locate the thousands place

Underline the digit to the right of the thousands place

The digit to the right of the thousands place is Since is greater than or equal to 5, round up by adding to the Then replace all digits to the right of the thousands place with zeros

So 29,504 rounded to the nearest thousand is 30,000

Notice that in partⓑ, when we add 1 thousand to the 9 thousands, the total is 10 thousands We regroup this as 1 ten thousand and0 thousands We add the 1ten thousand to the 3 ten thousands and put a 0in the thousands place

TRY IT : :1.19 Round to the nearest thousand: 63,921.

TRY IT : :1.20 Round to the nearest thousand: 156,437.

MEDIA : :ACCESS ADDITIONAL ONLINE RESOURCES

• Determine Place Value (http://www.openstaxcollege.org/l/24detplaceval)

Practice Makes Perfect

Identify Counting Numbers and Whole Numbers

In the following exercises, determine which of the following numbers areⓐcounting numbersⓑwhole numbers.

1. 0, 23, 5, 8.1, 125 2. 0, 710, 3, 20.5, 300 3. 0, 49, 3.9, 50, 221 4. 0, 35, 10, 303, 422.6

Model Whole Numbers

In the following exercises, use place value notation to find the value of the number modeled by the base-10blocks.

5. 6. 7.

8.

Identify the Place Value of a Digit

In the following exercises, find the place value of the given digits.

9. 579,601

ⓐ9 ⓑ6 ⓒ0 ⓓ7

ⓔ5

10.398,127

ⓐ9 ⓑ3 ⓒ2 ⓓ8

ⓔ7

11. 56,804,379

ⓐ8 ⓑ6 ⓒ4 ⓓ7

ⓔ0 12. 78,320,465

ⓐ8 ⓑ4 ⓒ2 ⓓ6

ⓔ7

Use Place Value to Name Whole Numbers

In the following exercises, name each number in words.

13. 1,078 14.5,902 15. 364,510

16. 146,023 17. 5,846,103 18. 1,458,398

19. 37,889,005 20. 62,008,465 21.The height of Mount Ranier is

14,410 feet 22.The height of Mount Adams is

12,276 feet 23.hours.Seventy years is 613,200 24.One year is 525,600 minutes 25. The U.S Census estimate of

the population of Miami-Dade county was 2,617,176.

26.The population of Chicago was

2,718,782. 27.23,867,000There are projected to becollege and university students in the US in five years

28.About twelve years ago there were 20,665,415 registered automobiles in California

29. The population of China is expected to reach1,377,583,156 in 2016.

30. The population of India is estimated at 1,267,401,849 as of July 1, 2014.

Use Place Value to Write Whole Numbers

In the following exercises, write each number as a whole number using digits.

31.four hundred twelve 32.two hundred fifty-three 33. thirty-five thousand, nine hundred seventy-five

34. sixty-one thousand, four

hundred fifteen 35.thousand, one hundred sixty-eleven million, forty-four seven

36.eighteen million, one hundred two thousand, seven hundred eighty-three

37. three billion, two hundred twenty-six million, five hundred twelve thousand, seventeen

38. eleven billion, four hundred seventy-one million, thirty-six thousand, one hundred six

39. The population of the world was estimated to be seven billion, one hundred seventy-three million people

40.The age of the solar system is estimated to be four billion, five hundred sixty-eight million years

41.Lake Tahoe has a capacity of thirty-nine trillion gallons of water

In the following exercises, round to the indicated place value.

43.Round to the nearest ten:

ⓐ 386 ⓑ 2,931

44.Round to the nearest ten:

ⓐ 792 ⓑ 5,647

45. Round to the nearest hundred:

ⓐ 13,748 ⓑ 391,794 46.Round to the nearest hundred:

ⓐ 28,166 ⓑ 481,628

47.Round to the nearest ten:

ⓐ 1,492 ⓑ 1,497

48. Round to the nearest thousand:

ⓐ 2,391 ⓑ 2,795 49. Round to the nearest

hundred:

ⓐ 63,994 ⓑ 63,949

50. Round to the nearest thousand:

ⓐ 163,584 ⓑ 163,246

Everyday Math

51.Writing a CheckJorge bought a car for $24,493. He paid for the car with a check Write the purchase price in words

52.Writing a CheckMarissa’s kitchen remodeling cost $18,549. She wrote a check to the contractor Write the amount paid in words

53. Buying a Car Jorge bought a car for $24,493. Round the price to the nearest:

ⓐten dollars ⓑhundred dollars

ⓒthousand dollars ⓓten-thousand dollars

54. Remodeling a Kitchen Marissa’s kitchen remodeling cost $18,549. Round the cost to the nearest:

ⓐten dollars ⓑhundred dollars

ⓒthousand dollars ⓓten-thousand dollars 55. Population The population of China was

1,355,692,544 in 2014. Round the population to the nearest:

ⓐbillion people ⓑhundred-million people

ⓒmillion people

56.Astronomy The average distance between Earth and the sun is 149,597,888 kilometers Round the distance to the nearest:

ⓐhundred-million kilometers

ⓑten-million kilometers ⓒmillion kilometers

Writing Exercises

57.In your own words, explain the difference between

the counting numbers and the whole numbers 58.helps to round numbers.Give an example from your everyday life where it

Self Check

ⓐAfter completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

can continue to use them What did you to become confident of your ability to these things? Be specific.

…with some help This must be addressed quickly because topics you not master become potholes in your road to success. In math, every topic builds upon previous work It is important to make sure you have a strong foundation before you move on. Who can you ask for help? Your fellow classmates and instructor are good resources Is there a place on campus where math tutors are available? Can your study skills be improved?

Add Whole Numbers Learning Objectives

By the end of this section, you will be able to:

Use addition notation

Model addition of whole numbers Add whole numbers without models Translate word phrases to math notation Add whole numbers in applications

Be Prepared!

Before you get started, take this readiness quiz

1 What is the number modeled by the base-10 blocks?

If you missed this problem, reviewExample 1.2

2 Write the number three hundred forty-two thousand six using digits? If you missed this problem, reviewExample 1.6

Use Addition Notation

A college student has a part-time job Last week he worked 3 hours on Monday and 4 hours on Friday To find the total number of hours he worked last week, he added 3 and 4.

The operation of addition combines numbers to get asum The notation we use to find the sum of 3and 4is: 3 + 4

We read this asthree plus fourand the result is the sum of three and four The numbers 3 and 4 are called the addends A math statement that includes numbers and operations is called an expression

Addition Notation

To describe addition, we can use symbols and words

Operation Notation Expression Read as Result

Addition + 3 + 4 three plus four the sum of 3 and 4 EXAMPLE 1.11

Translate from math notation to words:

ⓐ 7 + 1 ⓑ 12 + 14

Solution

ⓐThe expression consists of a plus symbol connecting the addends and We read this asseven plus one The result isthe sum of seven and one

The result is thesum of twelve and fourteen

TRY IT : :1.21 Translate from math notation to words:

ⓐ 8 + 4 ⓑ 18 + 11 TRY IT : :1.22 Translate from math notation to words:

ⓐ 21 + 16 ⓑ 100 + 200

Model Addition of Whole Numbers

Addition is really just counting We will model addition with base-10 blocks Remember, a block represents 1 and a rod represents 10. Let’s start by modeling the addition expression we just considered, 3 + 4.

Each addend is less than 10, so we can use ones blocks

We start by modeling the first number with blocks

Then we model the second number with blocks

Count the total number of blocks

There are 7 blocks in all We use an equal sign (=) to show the sum A math sentence that shows that two expressions are equal is called an equation We have shown that 3 + = 7.

MANIPULATIVE MATHEMATICS

Doing the Manipulative Mathematics activity “Model Addition of Whole Numbers” will help you develop a better understanding of adding whole numbers

EXAMPLE 1.12

Model the addition 2 + 6.

Solution

2 + 6 means the sum of 2 and 6

Model the first number with blocks Model the second number with blocks

Count the total number of blocks

There are 8 blocks in all, so 2 + = 8.

TRY IT : :1.23 Model: 3 + 6.

TRY IT : :1.24 Model: 5 + 1.

When the result is 10 or more ones blocks, we will exchange the 10 blocks for one rod EXAMPLE 1.13

Model the addition 5 + 8.

Solution

5 + 8 means the sum of 5 and 8.

Each addend is less than 10, se we can use ones blocks Model the first number with blocks

Model the second number with blocks

Count the result There are more than 10 blocks so we exchange 10 ones blocks for tens rod

Now we have ten and ones, which is 13 + = 13

Notice that we can describe the models as ones blocks and tens rods, or we can simply sayonesandtens From now on, we will use the shorter version but keep in mind that they mean the same thing

TRY IT : :1.25 Model the addition: 5 + 7. TRY IT : :1.26 Model the addition: 6 + 8.

Next we will model adding two digit numbers EXAMPLE 1.14

Model the addition: 17 + 26.

Solution

Model the 17 ten and ones Model the 26 tens and ones Combine tens and 13 ones

Exchange 10 ones for ten tens and ones40 + = 43 We have shown that 17 + 26 = 43

TRY IT : :1.27 Model each addition: 15 + 27.

TRY IT : :1.28 Model each addition: 16 + 29.

Add Whole Numbers Without Models

Now that we have used models to add numbers, we can move on to adding without models Before we that, make sure you know all the one digit addition facts You will need to use these number facts when you add larger numbers Imagine filling inTable 1.11by adding each row number along the left side to each column number across the top Make sure that you get each sum shown If you have trouble, model it It is important that you memorize any number facts you not already know so that you can quickly and reliably use the number facts when you add larger numbers

+ 0 1 2 3 4 5 6 7 8 9

0

1 10

2 10 11

3 10 11 12

4 10 11 12 13

5 10 11 12 13 14

6 10 11 12 13 14 15

7 10 11 12 13 14 15 16

8 10 11 12 13 14 15 16 17

9 10 11 12 13 14 15 16 17 18

Table 1.11

Did you notice what happens when you add zero to a number? The sum of any number and zero is the number itself We call this the Identity Property of Addition Zero is called the additive identity

Identity Property of Addition

0 + a = a EXAMPLE 1.15

Find each sum:

ⓐ 0 + 11 ⓑ 42 + 0

Solution

ⓐThe first addend is zero The sum of any number and zero is the number 0 + 11 = 11

ⓑThe second addend is zero The sum of any number and zero is the number 42 + = 42

TRY IT : :1.29 Find each sum:

ⓐ0 + 19 ⓑ 39 + 0 TRY IT : :1.30 Find each sum:

ⓐ 0 + 24 ⓑ 57 + 0 Look at the pairs of sums

2 + = 5 3 + = 5 4 + = 11 7 + = 11 8 + = 17 9 + = 17

Notice that when the order of the addends is reversed, the sum does not change This property is called the Commutative Property of Addition, which states that changing the order of the addends does not change their sum

Commutative Property of Addition

Changing the order of the addends a and b does not change their sum

a + b = b + a

EXAMPLE 1.16 Add:

ⓐ 8 + 7 ⓑ 7 + 8

Solution

ⓐ

Add 8 + 7

ⓑ

Add 7 + 8

15

Did you notice that changing the order of the addends did not change their sum? We could have immediately known the sum from partⓑjust by recognizing that the addends were the same as in partⓑ, but in the reverse order As a result, both sums are the same

TRY IT : :1.31 Add: 9 + 7 and 7 + 9. TRY IT : :1.32 Add: 8 + 6 and 6 + 8.

EXAMPLE 1.17 Add: 28 + 61.

Solution

To add numbers with more than one digit, it is often easier to write the numbers vertically in columns

Write the numbers so the ones and tens digits line up vertically +6128

Then add the digits in each place value Add the ones: 8 + = 9

Add the tens: 2 + = 8

28 +61 89

TRY IT : :1.33 Add: 32 + 54.

TRY IT : :1.34 Add: 25 + 74.

In the previous example, the sum of the ones and the sum of the tens were both less than 10. But what happens if the sum is 10 or more? Let’s use our base-10model to find out.Figure 1.10shows the addition of 17 and 26 again

Figure 1.10

When we add the ones, 7 + 6, we get 13 ones Because we have more than 10 ones, we can exchange 10 of the ones for 1 ten Now we have 4tens and 3 ones Without using the model, we show this as a small red 1 above the digits in the tens place

EXAMPLE 1.18 Add: 43 + 69.

Solution

Write the numbers so the digits line up vertically +6943

Add the digits in each place Add the ones: 3 + = 12

Write the 2 in the ones place in the sum Add the 1 ten to the tens place

413 +69 2

Now add the tens: 1 + + = 11

Write the 11 in the sum

413 +69 112

TRY IT : :1.35 Add: 35 + 98. TRY IT : :1.36 Add: 72 + 89.

EXAMPLE 1.19 Add: 324 + 586.

HOW TO : :ADD WHOLE NUMBERS

Write the numbers so each place value lines up vertically

Add the digits in each place value Work from right to left starting with the ones place If a sum in a place value is more than 9, carry to the next place value

Continue adding each place value from right to left, adding each place value and carrying if needed

Solution

Write the numbers so the digits line up vertically Add the digits in each place value

Add the ones: 4 + = 10

Write the 0 in the ones place in the sum and carry the 1 ten to the tens place Add the tens: 1 + + = 11

Write the 1 in the tens place in the sum and carry the 1 hundred to the hundreds Add the hundreds: 1 + + = 9

Write the 9 in the hundreds place

TRY IT : :1.37 Add: 456 + 376. TRY IT : :1.38 Add: 269 + 578.

EXAMPLE 1.20 Add: 1,683 + 479.

Solution

Write the numbers so the digits line up vertically + 4791,683

Add the digits in each place value Add the ones: 3 + = 12.

Write the 2 in the ones place of the sum and carry the 1 ten to the tens place

1,6813 + 479

2

Add the tens: 1 + + = 16

Write the 6 in the tens place and carry the 1 hundred to the hundreds place

1,61813 + 479

62

Add the hundreds: 1 + + = 11

Write the 1 in the hundreds place and carry the 1 thousand to the thousands place

1,61813 + 479

162

Add the thousands 1 + = 2

Write the 2 in the thousands place of the sum

1,61813 + 479

TRY IT : :1.39 Add: 4,597 + 685.

TRY IT : :1.40 Add: 5,837 + 695.

EXAMPLE 1.21

Add: 21,357 + 861 + 8,596.

Solution

Write the numbers so the place values line up vertically

21,357 861 + 8,596 _

Add the digits in each place value Add the ones: 7 + + = 14

Write the 4 in the ones place of the sum and carry the 1 to the tens place

21,3517 861 + 8,596 _ 4

Add the tens: 1 + + + = 21

Write the 1 in the tens place and carry the 2 to the hundreds place

21,32517 861 + 8,596 _ 14

Add the hundreds: 2 + + + = 18

Write the 8 in the hundreds place and carry the 1 to the thousands place

21,132517 861 + 8,596 _ 814

Add the thousands 1 + + = 10

Write the 0 in the thousands place and carry the 1 to the ten thousands place

211,132517 861 + 8,596 _ 0814

Add the ten-thousands 1 + = 3

Write the 3 in the ten thousands place in the sum

211,132517 861 + 8,596 _ 30,814

TRY IT : :1.42 Add: 53,762 + 196 + 7,458.

Translate Word Phrases to Math Notation

Earlier in this section, we translated math notation into words Now we’ll reverse the process We’ll translate word phrases into math notation Some of the word phrases that indicate addition are listed inTable 1.20

Operation Words Example Expression

Addition plus sum

increased by more than total of added to

1 plus 2

the sum of 3 and 4 5 increased by 6 8 more than 7

the total of 9 and 5 6 added to 4

1 + 2 3 + 4 5 + 6 7 + 8 9 + 5 4 + 6

Table 1.20

EXAMPLE 1.22

Translate and simplify: the sum of 19 and 23.

Solution

The wordsumtells us to add The wordsof 19 and 23 tell us the addends

The sum of 19 and 23

Translate 19 + 23 Add 42

The sum of 19 and 23 is 42.

TRY IT : :1.43 Translate and simplify: the sum of 17 and 26.

TRY IT : :1.44 Translate and simplify: the sum of 28 and 14.

EXAMPLE 1.23

Translate and simplify: 28 increased by 31.

Solution

The wordsincreased bytell us to add The numbers given are the addends 28 increased by 31.

Translate 28 + 31 Add 59

TRY IT : :1.46 Translate and simplify: 37 increased by 69.

Add Whole Numbers in Applications

Now that we have practiced adding whole numbers, let’s use what we’ve learned to solve real-world problems We’ll start by outlining a plan First, we need to read the problem to determine what we are looking for Then we write a word phrase that gives the information to find it Next we translate the word phrase into math notation and then simplify Finally, we write a sentence to answer the question

EXAMPLE 1.24

Hao earned grades of 87, 93, 68, 95, and 89 on the five tests of the semester What is the total number of points he earned on the five tests?

Solution

We are asked to find the total number of points on the tests

Write a phrase the sum of points on the tests Translate to math notation 87 + 93 + 68 + 95 + 89 Then we simplify by adding

Since there are several numbers, we will write them vertically

8

3

7 93 68 95 +89 432

Write a sentence to answer the question Hao earned a total of 432 points

Notice that we addedpoints, so the sum is 432 points It is important to include the appropriate units in all answers to applications problems

TRY IT : :1.47

Mark is training for a bicycle race Last week he rode 18 miles on Monday, 15 miles on Wednesday, 26 miles on Friday, 49 miles on Saturday, and 32 miles on Sunday What is the total number of miles he rode last week? TRY IT : :1.48

Lincoln Middle School has three grades The number of students in each grade is230, 165, and 325. What is the total number of students?

Some application problems involve shapes For example, a person might need to know the distance around a garden to put up a fence or around a picture to frame it The perimeter is the distance around a geometric figure The perimeter of a figure is the sum of the lengths of its sides

EXAMPLE 1.25

Solution

We are asked to find the perimeter

Write a phrase the sum of the sides Translate to math notation 4 + + + + + 9 Simplify by adding 26

Write a sentence to answer the question

We added feet, so the sum is 26 feet The perimeter of the patio is 26 feet

TRY IT : :1.49 Find the perimeter of each figure All lengths are in inches

TRY IT : :1.50 Find the perimeter of each figure All lengths are in inches

MEDIA : :ACCESS ADDITIONAL ONLINE RESOURCES

• Adding Two-Digit Numbers with base-10 blocks (http://www.openstaxcollege.org/l/24add2blocks)

• Adding Three-Digit Numbers with base-10 blocks (http://www.openstaxcollege.org/l/24add3blocks)

Practice Makes Perfect

Use Addition Notation

In the following exercises, translate the following from math expressions to words.

59. 5 + 2 60. 6 + 3 61. 13 + 18

62. 15 + 16 63.214 + 642 64. 438 + 113

Model Addition of Whole Numbers

In the following exercises, model the addition.

65. 2 + 4 66. 5 + 3 67. 8 + 4

68. 5 + 9 69. 14 + 75 70. 15 + 63

71. 16 + 25 72.14 + 27

Add Whole Numbers

In the following exercises, fill in the missing values in each chart.

73. 74. 75.

76. 77. 78.

In the following exercises, add.

79.

ⓐ 0 + 13 ⓑ 13 + 0

80.

ⓐ 0 + 5,280

ⓑ 5,280 + 0

81.

ⓐ 8 + 3 ⓑ 3 + 8 82.

ⓐ 7 + 5 ⓑ 5 + 7

83.45 + 33 84. 37 + 22

85. 71 + 28 86.43 + 53 87. 26 + 59

88. 38 + 17 89.64 + 78 90. 92 + 39

91. 168 + 325 92. 247 + 149 93. 584 + 277

94. 175 + 648 95.832 + 199 96. 775 + 369

97. 6,358 + 492 98.9,184 + 578 99. 3,740 + 18,593

100. 6,118 + 15,990 101. 485,012 + 619,848 102. 368,911 + 857,289 103. 24,731 + 592 + 3,868 104.28,925 + 817 + 4,593 105. 8,015 + 76,946 + 16,570 106. 6,291 + 54,107 + 28,635

Translate Word Phrases to Math Notation

In the following exercises, translate each phrase into math notation and then simplify.

107.the sum of 13 and 18 108.the sum of 12 and 19 109.the sum of 90 and 65 110.the sum of 70 and 38 111.33 increased by 49 112. 68 increased by 25 113. 250 more than 599 114. 115more than 286 115.the total of 628 and 77 116.the total of 593 and 79 117.1,482 added to 915 118. 2,719 added to 682

Add Whole Numbers in Applications

In the following exercises, solve the problem.

119. Home remodeling Sophia remodeled her kitchen and bought a new range, microwave, and dishwasher The range cost $1,100, the microwave cost $250, and the dishwasher cost $525. What was the total cost of these three appliances?

120. Sports equipment Aiden bought a baseball bat, helmet, and glove The bat cost $299, the helmet cost $35, and the glove cost $68. What was the total cost of Aiden’s sports equipment?

19 floral arrangements on Monday,

12 on Tuesday, 23 on

Wednesday, 29 on Thursday, and 44 on Friday What was the total number of floral arrangements Chloe made?

7

number of square feet in each

room is

238, 120, 156, 196, 100, 132, and 225. What is the total number of square feet in all 7 rooms?

men were

175, 192, 148, 169, 205, 181, and 225 pounds What was the total weight of the seven men?

125. Salary Last year Natalie’s salary was $82,572. Two years ago, her salary was $79,316, and three years ago it was

$75,298. What is the total amount of Natalie’s salary for the past three years?

126. Home sales Emma is a realtor Last month, she sold three houses The selling prices of the

houses were

$292,540, $505,875, and $423,699. What was the total of the three selling prices?

In the following exercises, find the perimeter of each figure.

127. 128. 129.

130. 131. 132.

133. 134.

Everyday Math

135.CaloriesPaulette had a grilled chicken salad, ranch dressing, and a 16-ounce drink for lunch On the restaurant’s nutrition chart, she saw that each item had the following number of calories:

Grilled chicken salad – 320 calories Ranch dressing – 170calories

16-ounce drink – 150calories

What was the total number of calories of Paulette’s

136. CaloriesFred had a grilled chicken sandwich, a small order of fries, and a 12-oz chocolate shake for dinner The restaurant’s nutrition chart lists the following calories for each item:

Grilled chicken sandwich – 420calories Small fries – 230 calories

12-oz chocolate shake – 580 calories

137.Test scoresA students needs a total of 400points on five tests to pass a course The student scored 82, 91, 75, 88, and 70. Did the student pass the course?

138. Elevators The maximum weight capacity of an elevator is 1150 pounds Six men are in the elevator Their weights are 210, 145, 183, 230, 159, and 164 pounds Is the total weight below the elevators’ maximum capacity?

Writing Exercises

139.How confident you feel about your knowledge of the addition facts? If you are not fully confident, what will you to improve your skills?

140.How have you used models to help you learn the addition facts?

Self Check

ⓐAfter completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

Subtract Whole Numbers Learning Objectives

By the end of this section, you will be able to:

Use subtraction notation

Model subtraction of whole numbers Subtract whole numbers

Translate word phrases to math notation Subtract whole numbers in applications

Be Prepared!

Before you get started, take this readiness quiz Model 3 + 4 using base-ten blocks

If you missed this problem, reviewExample 1.12 Add: 324 + 586.

If you missed this problem, reviewExample 1.20

Use Subtraction Notation

Suppose there are seven bananas in a bowl Elana uses three of them to make a smoothie How many bananas are left in the bowl? To answer the question, we subtract three from seven When we subtract, we take one number away from another to find thedifference The notation we use to subtract 3 from 7 is

7 − 3

We read 7 − 3 asseven minus threeand the result isthe difference of seven and three

Subtraction Notation

To describe subtraction, we can use symbols and words

Operation Notation Expression Read as Result

Subtraction − 7 − 3 seven minus three the difference of 7 and 3 EXAMPLE 1.26

Translate from math notation to words:ⓐ8 − 1 ⓑ 26 − 14

Solution

ⓐWe read this aseight minus one The result is the difference of eight and one

ⓑWe read this astwenty-six minus fourteen The resuilt is the difference of twenty-six and fourteen

TRY IT : :1.51 Translate from math notation to words:

ⓐ 12 − 4 ⓑ 29 − 11 TRY IT : :1.52 Translate from math notation to words:

ⓐ 11 − 2 ⓑ 29 − 12

Remember a block represents and a rod represents 10 Let’s start by modeling the subtraction expression we just considered, 7 − 3.

We start by modeling the first number,

Now take away the second number, We'll circle blocks to show that we are taking them away

Count the number of blocks remaining

There are ones blocks left We have shown that7 − = 4.

MANIPULATIVE MATHEMATICS

Doing the Manipulative Mathematics activity Model Subtraction of Whole Numbers will help you develop a better understanding of subtracting whole numbers

EXAMPLE 1.27

Model the subtraction: 8 − 2.

Solution

8 − 2 means the difference of and Model the first,

Take away the second number, Count the number of blocks remaining

There are ones blocks left We have shown that 8 − = 6

TRY IT : :1.53 Model: 9 − 6.

TRY IT : :1.54 Model: 6 − 1.

EXAMPLE 1.28

Model the first number, 13 We use ten and ones

Take away the second number, However, there are not ones, so we will exchange the ten for 10 ones

Now we can take away ones Count the blocks remaining

There are five ones left We have shown that 13 − = 5

As we did with addition, we can describe the models as ones blocks and tens rods, or we can simply say ones and tens TRY IT : :1.55 Model the subtraction: 12 − 7.

TRY IT : :1.56 Model the subtraction: 14 − 8. EXAMPLE 1.29

Model the subtraction: 43 − 26.

Solution

Because 43 − 26 means 43 take away 26, we begin by modeling the 43.

Now, we need to take away 26, which is 2 tens and 6 ones We cannot take away 6 ones from 3 ones So, we exchange 1 ten for 10 ones

Now we can take away 2tens and 6 ones

TRY IT : :1.57 Model the subtraction: 42 − 27.

TRY IT : :1.58 Model the subtraction: 45 − 29.

Subtract Whole Numbers

Addition and subtraction are inverse operations Addition undoes subtraction, and subtraction undoes addition

We know 7 − = 4 because 4 + = 7. Knowing all the addition number facts will help with subtraction Then we can check subtraction by adding In the examples above, our subtractions can be checked by addition

7 − = 4 because 4 + = 7 13 − = 5 because 5 + = 13 43 − 26 = 17 because 17 + 26 = 43 EXAMPLE 1.30

Subtract and then check by adding:

ⓐ 9 − 7 ⓑ 8 − 3.

Solution

ⓐ

9 − 7

Subtract from 2

Check with addition

2 + = ✓

ⓑ

8 − 3

Subtract from 5 Check with addition

5 + = ✓

TRY IT : :1.59 Subtract and then check by adding:

7 − 0

TRY IT : :1.60 Subtract and then check by adding:

6 − 2

To subtract numbers with more than one digit, it is usually easier to write the numbers vertically in columns just as we did for addition Align the digits by place value, and then subtract each column starting with the ones and then working to the left

EXAMPLE 1.31

Write the numbers so the ones and tens digits line up vertically −6189 Subtract the digits in each place value

Subtract the ones: 9 - = 8

Subtract the tens: 8 - = 2

89 −61 28

Check using addition

28 +61 89

Our answer is correct

TRY IT : :1.61 Subtract and then check by adding: 86 − 54. TRY IT : :1.62 Subtract and then check by adding: 99 − 74.

When we modeled subtracting 26 from 43, we exchanged 1ten for 10 ones When we this without the model, we say we borrow 1 from the tens place and add 10 to the ones place

EXAMPLE 1.32 Subtract: 43 − 26.

HOW TO : :FIND THE DIFFERENCE OF WHOLE NUMBERS Write the numbers so each place value lines up vertically

Subtract the digits in each place value Work from right to left starting with the ones place If the digit on top is less than the digit below, borrow as needed

Continue subtracting each place value from right to left, borrowing if needed Check by adding

Solution

Write the numbers so each place value lines up vertically

Subtract the ones We cannot subtract from 3, so we borrow ten This makes tens and 13 ones We write these numbers above each place and cross out the original digits

Now we can subtract the ones 13 − = 7. We write the in the ones place in the difference

Now we subtract the tens 3 − = 1. We write the in the tens place in the difference Check by adding

Our answer is correct

TRY IT : :1.63 Subtract and then check by adding: 93 − 58. TRY IT : :1.64 Subtract and then check by adding: 81 − 39.

EXAMPLE 1.33

Write the numbers so each place value lines up vertically Subtract the ones 7 − = 3.

Write the in the ones place in the difference Write the in the ones place in the difference Subtract the tens We cannot subtract from so we borrow hundred and add 10 tens to the tens we had This makes a total of 10 tens We write 10 above the tens place and cross out the Then we cross out the in the hundreds place and write above it

Now we subtract the tens 10 − = 4. We write the in the tens place in the difference Finally, subtract the hundreds There is no digit in the hundreds place in the bottom number so we can imagine a in that place Since 1 − = 1, we write in the hundreds place in the difference

Check by adding

Our answer is correct

TRY IT : :1.65 Subtract and then check by adding: 439 − 52.

TRY IT : :1.66 Subtract and then check by adding: 318 − 75.

EXAMPLE 1.34

Solution

Write the numbers so each place value lines up vertically

Subtract the ones We cannot subtract from 0, so we borrow ten and add 10 ones to the 10 ones we had This makes 10 ones We write a above the tens place and cross out the We write the 10 above the ones place and cross out the Now we can subtract the ones

10 − = 4.

Write the in the ones place of the difference

Subtract the tens We cannot subtract from 0, so we borrow hundred and add 10 tens to the tens we had, which gives us 10 tens Write above the hundreds place and cross out the Write 10 above the tens place

Now we can subtract the tens 10 − = 2

Subtract the hundreds place 8 − = 3 Write the in the hundreds place in the difference Check by adding

Our answer is correct

TRY IT : :1.67 Subtract and then check by adding: 832 − 376. TRY IT : :1.68 Subtract and then check by adding: 847 − 578.

EXAMPLE 1.35

Write the numbers so each place values line up vertically

Subtract the ones Since we cannot subtract from 2, borrow ten and add 10 ones to the ones to make 12 ones Write above the tens place and cross out the Write 12 above the ones place and cross out the

Now we can subtract the ones 12 − = 3

Write in the ones place in the difference

Subtract the tens Since we cannot subtract from 5, borrow hundred and add 10 tens to the tens to make 15 tens Write above the hundreds place and cross out the Write 15 above the tens place

Now we can subtract the tens 15 − = 8

Write in the tens place in the difference

Now we can subtract the hundreds

Write in the hundreds place in the difference

Subtract the thousands There is no digit in the thousands place of the bottom number, so we imagine a 1 − = 1. Write in the thousands place of the difference

Check by adding

11, 61813 + 479

2, 162 ✓ Our answer is correct

TRY IT : :1.69 Subtract and then check by adding: 4,585 − 697.

TRY IT : :1.70 Subtract and then check by adding: 5,637 − 899.

inTable 1.34

Operation Word Phrase Example Expression

Subtraction minus 5 minus 1 5 − 1

difference the difference of 9 and 4 9 − 4

decreased by 7 decreased by 3 7 − 3

less than 5 less than 8 8 − 5

subtracted from 1 subtracted from 6 6 − 1

Table 1.34

EXAMPLE 1.36

Translate and then simplify:

ⓐthe difference of 13 and 8 ⓑsubtract 24 from 43

Solution

ⓐ

The worddifferencetells us to subtract the two numbers The numbers stay in the same order as in the phrase

the difference of 13 and Translate 13 − 8

Simplify ⓑ

The wordssubtract fromtells us to take the second number away from the first We must be careful to get the order correct

subtract 24 from 43 Translate 43 − 24

Simplify 19

TRY IT : :1.71 Translate and simplify:

ⓐthe difference of 14 and 9 ⓑsubtract 21 from 37 TRY IT : :1.72 Translate and simplify:

ⓐ 11 decreased by 6 ⓑ 18 less than 67

Subtract Whole Numbers in Applications

The temperature in Chicago one morning was 73 degrees Fahrenheit A cold front arrived and by noon the temperature was 27 degrees Fahrenheit What was the difference between the temperature in the morning and the temperature at noon?

Solution

We are asked to find the difference between the morning temperature and the noon temperature

Write a phrase the difference of 73 and 27 Translate to math notation.Differencetells us to

subtract 73 − 27

Then we the subtraction

Write a sentence to answer the question The difference in temperatures was 46 degreesFahrenheit.

TRY IT : :1.73

The high temperature on June 1st in Boston was 77 degrees Fahrenheit, and the low temperature was 58 degrees Fahrenheit What was the difference between the high and low temperatures?

TRY IT : :1.74

The weather forecast for June 2 in St Louis predicts a high temperature of 90 degrees Fahrenheit and a low of 73 degrees Fahrenheit What is the difference between the predicted high and low temperatures?

EXAMPLE 1.38

A washing machine is on sale for $399. Its regular price is $588. What is the difference between the regular price and the sale price?

Solution

We are asked to find the difference between the regular price and the sale price

Write a phrase the difference between 588 and 399 Translate to math notation 588 − 399

Subtract

Write a sentence to answer the

question The difference between the regular price and the sale price is$189

TRY IT : :1.75

TRY IT : :1.76

A patio set is on sale for $149. Its regular price is $285. What is the difference between the regular price and the sale price?

MEDIA : :ACCESS ADDITIONAL ONLINE RESOURCES

• Model subtraction of two-digit whole numbers (http://www.openstaxcollege.org/l/24sub2dignum)

• Model subtraction of three-digit whole numbers (http://www.openstaxcollege.org/l/24sub3dignum)

Practice Makes Perfect

Use Subtraction Notation

In the following exercises, translate from math notation to words.

141. 15 − 9 142. 18 − 16 143. 42 − 35

144. 83 − 64 145. 675 − 350 146. 790 − 525

Model Subtraction of Whole Numbers

In the following exercises, model the subtraction.

147. 5 − 2 148. 8 − 4 149. 6 − 3

150. 7 − 5 151.18 − 5 152. 19 − 8

153. 17 − 8 154. 17 − 9 155. 35 − 13

156. 32 − 11 157. 61 − 47 158. 55 − 36

Subtract Whole Numbers

In the following exercises, subtract and then check by adding.

159. 9 − 4 160.9 − 3 161. 8 − 0

162. 2 − 0 163. 38 − 16 164. 45 − 21

165. 85 − 52 166. 99 − 47 167. 493 − 370

168. 268 − 106 169. 5,946 − 4,625 170. 7,775 − 3,251

171. 75 − 47 172. 63 − 59 173. 461 − 239

174. 486 − 257 175. 525 − 179 176. 542 − 288

177. 6,318 − 2,799 178. 8,153 − 3,978 179. 2,150 − 964

180. 4,245 − 899 181.43,650 − 8,982 182. 35,162 − 7,885

Translate Word Phrases to Algebraic Expressions

In the following exercises, translate and simplify.

183.The difference of 10 and 3 184.The difference of 12 and 8 185.The difference of 15 and 4 186.The difference of 18 and 7 187.Subtract 6from 9 188.Subtract 8 from 9

189.Subtract 28 from 75 190.Subtract 59 from 81 191. 45 decreased by 20 192. 37 decreased by 24 193. 92 decreased by 67 194. 75 decreased by 49 195. 12 less than 16 196. 15 less than 19 197. 38 less than 61

198. 47 less than 62

Mixed Practice

In the following exercises, simplify.

199. 76−47 200. 91 − 53 201. 256 − 184

202. 305 − 262 203.719 + 341 204. 647 + 528

205. 2,015 − 1,993 206. 2,020 − 1,984

In the following exercises, translate and simplify.

207.Seventy more than thirty-five 208.Sixty more than ninety-three 209. 13 less than 41 210. 28 less than 36 211. The difference of 100 and

76 212.945The difference of 1,000 and

Subtract Whole Numbers in Applications

In the following exercises, solve.

213. Temperature The high temperature on June 2 in Las Vegas was 80 degrees and the low temperature was 63 degrees What was the difference between the high and low temperatures?

214. Temperature The high temperature on June 1 in Phoenix was 97 degrees and the low was 73 degrees What was the difference between the high and low temperatures?

215.Class sizeOlivia’s third grade class has 35 children Last year, her second grade class had 22 children What is the difference between the number of children in Olivia’s third grade class and her second grade class?

216. Class size There are 82 students in the school band and

46 in the school orchestra What is the difference between the number of students in the band and the orchestra?

217.ShoppingA mountain bike is on sale for $399. Its regular price is $650. What is the difference between the regular price and the sale price?

218.ShoppingA mattress set is on sale for $755. Its regular price is $1,600. What is the difference between the regular price and the sale price?

219.SavingsJohn wants to buy a laptop that costs $840. He has $685 in his savings account How much more does he need to save in order to buy the laptop?

220.BankingMason had $1,125 in his checking account He spent

$892. How much money does he have left?

Everyday Math

221.Road tripNoah was driving from Philadelphia to Cincinnati, a distance of 502 miles He drove 115 miles, stopped for gas, and then drove another 230 miles before lunch How many more miles did he have to travel?

222.Test Scores Sara needs 350 points to pass her course She scored 75, 50, 70, and 80 on her first four tests How many more points does Sara need to pass the course?

Writing Exercises

ⓐAfter completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

1.4 Multiply Whole Numbers Learning Objectives

By the end of this section, you will be able to:

Use multiplication notation

Model multiplication of whole numbers Multiply whole numbers

Translate word phrases to math notation Multiply whole numbers in applications

Be Prepared!

Before you get started, take this readiness quiz Add: 1,683 + 479.

If you missed this problem, reviewExample 1.21 Subtract: 605 − 321.

If you missed this problem, reviewExample 1.33

Use Multiplication Notation

Suppose you were asked to count all these pennies shown inFigure 1.11

Figure 1.11

Would you count the pennies individually? Or would you count the number of pennies in each row and add that number 3 times

8 + + 8

Multiplication is a way to represent repeated addition So instead of adding 8 three times, we could write a multiplication expression

3 × 8

We call each number being multiplied a factor and the result theproduct We read 3 × 8 asthree times eight, and the result asthe product of three and eight

There are several symbols that represent multiplication These include the symbol × as well as the dot, ·, and parentheses ( ).

Operation Symbols for Multiplication

To describe multiplication, we can use symbols and words

Operation Notation Expression Read as Result

Multiplication × · ( )

3 × 8 3 · 8 3(8)

Translate from math notation to words:

ⓐ 7 × 6 ⓑ 12 · 14 ⓒ 6(13)

Solution

ⓐWe read this asseven times six and the result is the product of seven and six

ⓑWe read this astwelve times fourteen and the result is the product of twelve and fourteen

ⓒWe read this assix times thirteen and the result is the product of six and thirteen

TRY IT : :1.77 Translate from math notation to words:

ⓐ 8 × 7 ⓑ 18 · 11

TRY IT : :1.78 Translate from math notation to words:

ⓐ (13)(7) ⓑ 5(16)

Model Multiplication of Whole Numbers

There are many ways to model multiplication Unlike in the previous sections where we used base-10 blocks, here we will use counters to help us understand the meaning of multiplication A counter is any object that can be used for counting We will use round blue counters

EXAMPLE 1.40 Model: 3 × 8.

Solution

To model the product 3 × 8, we’ll start with a row of 8 counters

The other factor is 3, so we’ll make 3 rows of 8counters

Now we can count the result There are 24 counters in all 3 × = 24

If you look at the counters sideways, you’ll see that we could have also made 8 rows of 3 counters The product would have been the same We’ll get back to this idea later

TRY IT : :1.79 Model each multiplication: 4 × 6.

TRY IT : :1.80 Model each multiplication: 5 × 7.

Multiply Whole Numbers

facts you not already know so you will be ready to multiply larger numbers

× 0 1 2 3 4 5 6 7 8 9

0 0 0 0 0 0

1

2 10 12 14 16 18

3 12 15 18 21 24 27

4 12 16 20 24 28 32 36

5 10 15 20 25 30 35 40 45

6 12 18 24 30 36 42 48 54

7 14 21 28 35 42 49 56 63

8 16 24 32 40 48 56 64 72

9 18 27 36 45 54 63 72 81

Table 1.39

What happens when you multiply a number by zero? You can see that the product of any number and zero is zero This is called the Multiplication Property of Zero

Multiplication Property of Zero The product of any number and 0 is 0.

a · = 0

0 · a = 0 EXAMPLE 1.41

Multiply:

ⓐ 0 · 11 ⓑ (42)0

Solution

ⓐ 0 · 11

The product of any number and zero is zero 0

ⓑ (42)0

Multiplying by zero results in zero 0

TRY IT : :1.81 Find each product:

ⓐ 0 · 19 ⓑ (39)0 TRY IT : :1.82 Find each product:

1

Identity Property of Multiplication

The product of any number and 1 is the number

1 · a = a

a · = a

EXAMPLE 1.42 Multiply:

ⓐ (11)1 ⓑ 1 · 42

Solution

ⓐ (11)1

The product of any number and one is the number 11

ⓑ 1 · 42

Multiplying by one does not change the value 42

TRY IT : :1.83 Find each product:

ⓐ (19)1 ⓑ 1 · 39 TRY IT : :1.84 Find each product:

ⓐ(24)(1) ⓑ 1 × 57

Earlier in this chapter, we learned that the Commutative Property of Addition states that changing the order of addition does not change the sum We saw that 8 + = 17 is the same as 9 + = 17.

Is this also true for multiplication? Let’s look at a few pairs of factors 4 · = 28 7 · = 28 9 · = 63 7 · = 63 8 · = 72 9 · = 72

When the order of the factors is reversed, the product does not change This is called the Commutative Property of Multiplication

Commutative Property of Multiplication

Changing the order of the factors does not change their product

a · b = b · a

EXAMPLE 1.43 Multiply:

Solution

ⓐ 8 · 7

Multiply 56

ⓑ 7 · 8

Multiply 56

Changing the order of the factors does not change the product TRY IT : :1.85 Multiply:

ⓐ 9 · 6 ⓑ 6 · 9 TRY IT : :1.86 Multiply:

ⓐ 8 · 6 ⓑ 6 · 8

To multiply numbers with more than one digit, it is usually easier to write the numbers vertically in columns just as we did for addition and subtraction

27 ×3 _ We start by multiplying 3by 7.

3 × = 21

We write the 1in the ones place of the product We carry the 2 tens by writing 2 above the tens place

Then we multiply the 3 by the 2, and add the 2 above the tens place to the product So 3 × = 6, and 6 + = 8. Write the 8in the tens place of the product

The product is 81.

When we multiply two numbers with a different number of digits, it’s usually easier to write the smaller number on the bottom You could write it the other way, too, but this way is easier to work with

Write the numbers so the digits 5 and 4 line up vertically _× 415 Multiply 4 by the digit in the ones place of 15 ⋅ = 20.

Write 0 in the ones place of the product and carry the 2 tens 1

5 × 4 _

0

Multiply 4 by the digit in the tens place of 15 ⋅ = 4 Add the 2 tens we carried 4 + = 6

Write the 6 in the tens place of the product 1

5 × 4 _

60

TRY IT : :1.87 Multiply: 64 · 8. TRY IT : :1.88 Multiply: 57 · 6.

EXAMPLE 1.45 Multiply: 286 · 5.

Solution

Write the numbers so the digits 5 and 6 line up vertically _286× 5 Multiply 5 by the digit in the ones place of 286 ⋅ = 30.

Write the 0 in the ones place of the product and carry the 3 to the tens place.Multiply 5 by the digit in the tens place of 286 ⋅ = 40

2836 × 5 _

0

Add the 3 tens we carried to get 40 + = 43

Write the 3 in the tens place of the product and carry the to the hundreds place

24836 × 5 _

30

Multiply 5 by the digit in the hundreds place of 286 ⋅ = 10.

Add the 4 hundreds we carried to get 10 + = 14.

Write the 4 in the hundreds place of the product and the 1 to the thousands place

24836 × 5 _ 1,430

When we multiply by a number with two or more digits, we multiply by each of the digits separately, working from right to left Each separate product of the digits is called a partial product When we write partial products, we must make sure to line up the place values

EXAMPLE 1.46 Multiply: 62(87).

HOW TO : :MULTIPLY TWO WHOLE NUMBERS TO FIND THE PRODUCT Write the numbers so each place value lines up vertically Multiply the digits in each place value

◦ Work from right to left, starting with the ones place in the bottom number

▪ Multiply the bottom number by the ones digit in the top number, then by the tens digit, and so on

▪ If a product in a place value is more than 9, carry to the next place value ▪ Write the partial products, lining up the digits in the place values with the

numbers above

◦ Repeat for the tens place in the bottom number, the hundreds place, and so on ◦ Insert a zero as a placeholder with each additional partial product

Add the partial products Step

Step

Write the numbers so each place lines up vertically

Start by multiplying by 62 Multiply by the digit in the ones place of 62 7 ⋅ = 14. Write the in the ones place of the product and carry the to the tens place

Multiply by the digit in the tens place of 62 7 ⋅ = 42. Add the ten we carried 42 + = 43 Write the in the tens place of the product and the in the hundreds place

The first partial product is 434

Now, write a under the in the ones place of the next partial product as a placeholder since we now multiply the digit in the tens place of 87 by 62 Multiply by the digit in the ones place of 62

8 ⋅ = 16. Write the in the next place of the product, which is the tens place Carry the to the tens place

Multiply by 6, the digit in the tens place of 62, then add the ten we carried to get 49 Write the in the hundreds place of the product and the in the thousands place

The second partial product is 4960 Add the partial products

The product is 5,394.

TRY IT : :1.91 Multiply: 43(78). TRY IT : :1.92 Multiply: 64(59).

EXAMPLE 1.47 Multiply:

Solution

ⓐ 47 · 10

47 ×10 _ 00 470 _ 470

ⓑ 47 · 100

47 ×100 _ 00 000 4700 _ 4,700

When we multiplied 47 times 10, the product was 470. Notice that 10 has one zero, and we put one zero after 47 to get the product When we multiplied 47 times 100, the product was 4,700. Notice that 100 has two zeros and we put two zeros after 47 to get the product

Do you see the pattern? If we multiplied 47 times 10,000, which has four zeros, we would put four zeros after 47 to get the product 470,000.

TRY IT : :1.93 Multiply:

ⓐ 54 · 10 ⓑ 54 · 100. TRY IT : :1.94 Multiply:

ⓐ 75 · 10 ⓑ 75 · 100. EXAMPLE 1.48

Multiply: (354)(438).

Solution

There are three digits in the factors so there will be 3 partial products We not have to write the 0 as a placeholder as long as we write each partial product in the correct place

TRY IT : :1.95 Multiply: (265)(483).

TRY IT : :1.96 Multiply: (823)(794). EXAMPLE 1.49

There should be 3 partial products The second partial product will be the result of multiplying 896 by 0.

Notice that the second partial product of all zeros doesn’t really affect the result We can place a zero as a placeholder in the tens place and then proceed directly to multiplying by the 2 in the hundreds place, as shown

Multiply by 10, but insert only one zero as a placeholder in the tens place Multiply by 200, putting the 2 from the 12 · = 12 in the hundreds place

896 ×201 _

896 17920

180,096 TRY IT : :1.97 Multiply: (718)509.

TRY IT : :1.98 Multiply: (627)804.

When there are three or more factors, we multiply the first two and then multiply their product by the next factor For example:

to multiply 8 ⋅ ⋅ 2

first multiply 8 ⋅ 3 24 ⋅ 2

then multiply 24 ⋅ 2 48

Translate Word Phrases to Math Notation

Earlier in this section, we translated math notation into words Now we’ll reverse the process and translate word phrases into math notation Some of the words that indicate multiplication are given inTable 1.47

Operation Word Phrase Example Expression

Multiplication times product twice

3 times 8

the product of 3 and 8

twice 4

3 × 8, · 8, (3)(8), (3)8, or 3(8) 2 · 4

Table 1.47

EXAMPLE 1.50

Translate and simplify: the product of 12 and 27.

Solution

the product of 12 and 27 Translate 12 ⋅ 27

Multiply 324

TRY IT : :1.99 Translate and simplify the product of 13 and 28.

TRY IT : :1.100 Translate and simplify the product of 47 and 14.

EXAMPLE 1.51

Translate and simplify: twice two hundred eleven

Solution

The wordtwicetells us to multiply by 2.

twice two hundred eleven Translate 2(211)

Multiply 422

TRY IT : :1.101 Translate and simplify: twice one hundred sixty-seven.

TRY IT : :1.102 Translate and simplify: twice two hundred fifty-eight.

Multiply Whole Numbers in Applications

We will use the same strategy we used previously to solve applications of multiplication First, we need to determine what we are looking for Then we write a phrase that gives the information to find it We then translate the phrase into math notation and simplify to get the answer Finally, we write a sentence to answer the question

EXAMPLE 1.52

Humberto bought 4 sheets of stamps Each sheet had 20 stamps How many stamps did Humberto buy?

Solution

We are asked to find the total number of stamps

Write a phrase for the total the product of and 20 Translate to math notation 4 ⋅ 20

Multiply

Valia donated water for the snack bar at her son’s baseball game She brought 6 cases of water bottles Each case had 24 water bottles How many water bottles did Valia donate?

TRY IT : :1.104

Vanessa brought 8 packs of hot dogs to a family reunion Each pack has 10 hot dogs How many hot dogs did Vanessa bring?

EXAMPLE 1.53

When Rena cooks rice, she uses twice as much water as rice How much water does she need to cook 4 cups of rice?

Solution

We are asked to find how much water Rena needs

Write as a phrase twice as much as cups Translate to math notation 2 ⋅ 4

Multiply to simplify

Write a sentence to answer the question Rena needs cups of water for cups of rice

TRY IT : :1.105

Erin is planning her flower garden She wants to plant twice as many dahlias as sunflowers If she plants 14 sunflowers, how many dahlias does she need?

TRY IT : :1.106

A college choir has twice as many women as men There are 18 men in the choir How many women are in the choir?

EXAMPLE 1.54

Van is planning to build a patio He will have 8 rows of tiles, with 14 tiles in each row How many tiles does he need for the patio?

Solution

We are asked to find the total number of tiles

Write a phrase the product of and 14 Translate to math notation 8 ⋅ 14

Multiply to simplify

1

3

4 ×8 _ 112

TRY IT : :1.107

Jane is tiling her living room floor She will need 16 rows of tile, with 20 tiles in each row How many tiles does she need for the living room floor?

TRY IT : :1.108

Yousef is putting shingles on his garage roof He will need 24 rows of shingles, with 45 shingles in each row How many shingles does he need for the garage roof?

If we want to know the size of a wall that needs to be painted or a floor that needs to be carpeted, we will need to find itsarea The area is a measure of the amount of surface that is covered by the shape Area is measured in square units We often use square inches, square feet, square centimeters, or square miles to measure area A square centimeter is a square that is one centimeter (cm.) on a side A square inch is a square that is one inch on each side, and so on

For a rectangular figure, the area is the product of the length and the width.Figure 1.12shows a rectangular rug with a length of 2 feet and a width of 3feet Each square is 1 foot wide by 1 foot long, or 1 square foot The rug is made of

6 squares The area of the rug is 6 square feet

Figure 1.12 The area of a rectangle is the product of its length and its width, or 6 square feet

EXAMPLE 1.55

Jen’s kitchen ceiling is a rectangle that measures feet long by 12 feet wide What is the area of Jen’s kitchen ceiling?

Solution

We are asked to find the area of the kitchen ceiling

Write a phrase for the area the product of and 12 Translate to math notation 9 ⋅ 12

Multiply

112 ×9 _ 108

Answer with a sentence The area of Jen's kitchen ceiling is 108 square feet

TRY IT : :1.109

Zoila bought a rectangular rug The rug is feet long by feet wide What is the area of the rug? TRY IT : :1.110

• Multiplying Whole Numbers (http://www.openstaxcollege.org/l/24multwhlnum)

• Multiplication with Partial Products (http://www.openstaxcollege.org/l/24multpartprod)

Practice Makes Perfect

Use Multiplication Notation

In the following exercises, translate from math notation to words.

225. 4 × 7 226.8 × 6 227. 5 · 12

228. 3 · 9 229.(10)(25) 230. (20)(15)

231. 42(33) 232.39(64)

Model Multiplication of Whole Numbers

In the following exercises, model the multiplication.

233. 3 × 6 234.4 × 5 235. 5 × 9

236. 3 × 9

Multiply Whole Numbers

In the following exercises, fill in the missing values in each chart.

237. 238. 239.

240. 241. 242.

In the following exercises, multiply.

245. 0 · 15 246. 0 · 41 247. (99)0

248. (77)0 249. 1 · 43 250. 1 · 34

251. (28)1 252. (65)1 253. 1(240,055)

254. 1(189,206) 255.

ⓐ 7 · 6 ⓑ 6 · 7

256.

ⓐ 8 × 9 ⓑ 9 × 8

257. (79)(5) 258. (58)(4) 259. 275 · 6

260. 638 · 5 261. 3,421 × 7 262. 9,143 × 3

263. 52(38) 264. 37(45) 265. 96 · 73

266. 89 · 56 267.27 × 85 268. 53 × 98

269. 23 · 10 270.19 · 10 271. (100)(36)

272. (100)(25) 273. 1,000(88) 274. 1,000(46)

275. 50 × 1,000,000 276.30 × 1,000,000 277. 247 × 139

278. 156 × 328 279.586(721) 280. 472(855)

281. 915 · 879 282.968 · 926 283. (104)(256)

284. (103)(497) 285. 348(705) 286. 485(602)

287. 2,719 × 543 288. 3,581 × 724

Translate Word Phrases to Math Notation

In the following exercises, translate and simplify.

295. ten times three hundred

seventy-five 296.five ten times two hundred

fifty-Mixed Practice

In the following exercises, simplify.

297. 38 × 37 298.86 × 29 299. 415 − 267

300. 341 − 285 301. 6,251 + 4,749 302. 3,816 + 8,184

303. (56)(204) 304. (77)(801) 305. 947 · 0

306. 947 + 0 307.15,382 + 1 308. 15,382 · 1

In the following exercises, translate and simplify.

309.the difference of 50 and 18 310.the difference of 90 and 66 311.twice 35

312.twice 140 313.20 more than 980 314.65 more than 325

315.the product of 12 and 875 316.the product of 15 and 905 317.subtract 74 from 89 318.subtract 45 from 99 319.the sum of 3,075 and 95 320.the sum of 6,308 and 724 321.366 less than 814 322.388 less than 925

Multiply Whole Numbers in Applications

In the following exercises, solve.

323.Party suppliesTim brought six-packs of soda to a club party How many cans of soda did Tim bring?

324.Sewing Kanisha is making a quilt She bought cards of buttons Each card had four buttons on it How many buttons did Kanisha buy?

325. Field trip Seven school busses let off their students in front of a museum in Washington, DC Each school bus had 44 students How many students were there?

326.GardeningKathryn bought flats of impatiens for her flower bed Each flat has 24 flowers How many flowers did Kathryn buy?

327. Charity Rey donated 15 twelve-packs of t-shirts to a homeless shelter How many t-shirts did he donate?

328. School There are 28 classrooms at Anna C Scott elementary school Each classroom has 26 student desks What is the total number of student desks?

329.Recipe Stephanie is making punch for a party The recipe calls for twice as much fruit juice as club soda If she uses 10 cups of club soda, how much fruit juice should she use?

330.GardeningHiroko is putting in a vegetable garden He wants to have twice as many lettuce plants as tomato plants If he buys 12 tomato plants, how many lettuce plants should he get?

331. Government The United States Senate has twice as many senators as there are states in the United States There are 50 states How many senators are there in the United States Senate? 332. Recipe Andrea is making

potato salad for a buffet luncheon The recipe says the number of servings of potato salad will be twice the number of pounds of potatoes If she buys 30 pounds of potatoes, how many servings of potato salad will there be?

333.PaintingJane is painting one wall of her living room The wall is rectangular, 13 feet wide by feet high What is the area of the wall?

with length 42 feet and width 34 feet What is the area of the meeting room?

garden is rectangular, with length 23 feet and width 28 feet What is the area of the garden?

dimensions of a rectangular basketball court must be 94 feet by 50 feet What is the area of the basketball court?

338. NCAA footballAccording to NCAA regulations, the dimensions of a rectangular football field must be 360 feet by 160 feet What is the area of the football field?

Everyday Math

339.Stock marketJavier owns 300 shares of stock in one company On Tuesday, the stock price rose $12 per share How much money did Javier’s portfolio gain?

340.SalaryCarlton got a $200 raise in each paycheck He gets paid 24 times a year How much higher is his new annual salary?

Writing Exercises

341.How confident you feel about your knowledge of the multiplication facts? If you are not fully confident, what will you to improve your skills?

342.How have you used models to help you learn the multiplication facts?

Self Check

ⓐAfter completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

1.5 Divide Whole Numbers Learning Objectives

By the end of this section, you will be able to:

Use division notation

Model division of whole numbers Divide whole numbers

Translate word phrases to math notation Divide whole numbers in applications

Be Prepared!

Before you get started, take this readiness quiz Multiply: 27 · 3.

If you missed this problem, reviewExample 1.44 Subtract: 43 − 26.

If you missed this problem, reviewExample 1.32

3 Multiply: 62(87).

If you missed this problem, reviewExample 1.45

Use Division Notation

So far we have explored addition, subtraction, and multiplication Now let’s consider division Suppose you have the 12 cookies inFigure 1.13and want to package them in bags with 4cookies in each bag How many bags would we need?

Figure 1.13

You might put 4 cookies in first bag, 4 in the second bag, and so on until you run out of cookies Doing it this way, you would fill 3 bags

In other words, starting with the 12 cookies, you would take away, or subtract, 4 cookies at a time Division is a way to represent repeated subtraction just as multiplication represents repeated addition

Instead of subtracting 4repeatedly, we can write

12 ÷ 4

We read this astwelve divided by fourand the result is thequotientof 12 and 4. The quotient is 3 because we can subtract 4 from 12 exactly 3 times We call the number being divided thedividendand the number dividing it the

divisor In this case, the dividend is 12 and the divisor is 4.

In the past you may have used the notation 4 12, but this division also can be written as 12 ÷ 4, 12/4, 124 In each case the 12 is the dividend and the 4 is the divisor

Operation Symbols for Division

Operation Notation Expression Read as Result

Division ÷

a b b a a/ b

12 ÷ 4 12

4 4 12 12 / 4

Twelve divided by four the quotient of 12 and 4

Division is performed on two numbers at a time When translating from math notation to English words, or English words to math notation, look for the wordsofandandto identify the numbers

EXAMPLE 1.56

Translate from math notation to words

ⓐ 64 ÷ 8 ⓑ 427 ⓒ4 28

Solution

ⓐWe read this assixty-four divided by eight and the result is the quotient of sixty-four and eight

ⓑWe read this asforty-two divided by seven and the result is the quotient of forty-two and seven

ⓒWe read this astwenty-eight divided by four and the result is the quotient of twenty-eight and four

TRY IT : :1.111 Translate from math notation to words:

ⓐ 84 ÷ 7 ⓑ 186 ⓒ8 24

TRY IT : :1.112 Translate from math notation to words:

ⓐ 72 ÷ 9 ⓑ 213 ⓒ 6 54

Model Division of Whole Numbers

As we did with multiplication, we will model division using counters The operation of division helps us organize items into equal groups as we start with the number of items in the dividend and subtract the number in the divisor repeatedly

MANIPULATIVE MATHEMATICS

Doing the Manipulative Mathematics activity Model Division of Whole Numbers will help you develop a better understanding of dividing whole numbers

EXAMPLE 1.57

Model the division: 24 ÷ 8.

Solution

The divisor tell us the number of counters we want in each group Form groups of 8 counters

Count the number of groups There are 3 groups 24 ÷ = 3

TRY IT : :1.113 Model: 24 ÷ 6. TRY IT : :1.114 Model: 42 ÷ 7.

Divide Whole Numbers

We said that addition and subtraction are inverse operations because one undoes the other Similarly, division is the inverse operation of multiplication We know 12 ÷ = 3 because 3 · = 12. Knowing all the multiplication number facts is very important when doing division

We check our answer to division by multiplying the quotient by the divisor to determine if it equals the dividend In

Example 1.57, we know 24 ÷ = 3 is correct because 3 · = 24. EXAMPLE 1.58

Divide Then check by multiplying.ⓐ 42 ÷ 6 ⓑ 729 ⓒ 7 63

Solution

ⓐ

42 ÷ 6

Divide 42 by 7 Check by multiplying

ⓑ

72 9

Divide 72 by 8 Check by multiplying

8 · 9 72 ✓

ⓒ

7 63

Divide 63 by 9 Check by multiplying

9 · 7 63 ✓

TRY IT : :1.115 Divide Then check by multiplying:

ⓐ 54 ÷ 6 ⓑ 279

TRY IT : :1.116 Divide Then check by multiplying:

ⓐ 36

9 ⓑ 8 40

What is the quotient when you divide a number by itself? 15

15 = because · 15 = 15

Dividing any number (except 0) by itself produces a quotient of 1. Also, any number divided by 1 produces a quotient of the number These two ideas are stated in the Division Properties of One

Division Properties of One

Any number (except 0) divided by itself is one a ÷ a = 1 Any number divided by one is the same number a ÷ = a Table 1.57

EXAMPLE 1.59

ⓐ 11 ÷ 11 ⓑ 191 ⓒ 1 7

Solution

ⓐ

11 ÷ 11

A number divided by itself is 1 Check by multiplying

1 · 11 11 ✓

ⓑ

19 1

A number divided by equals itself 19 Check by multiplying

19 · 1 19 ✓

ⓒ

1 7

A number divided by equals itself 7 Check by multiplying

7 · 1 7 ✓

TRY IT : :1.117 Divide Then check by multiplying:

ⓐ 14 ÷ 14 ⓑ 271

TRY IT : :1.118 Divide Then check by multiplying:

ⓐ 16 1 ⓑ 1 4

Suppose we have $0, and want to divide it among 3 people How much would each person get? Each person would get $0. Zero divided by any number is 0.

Now suppose that we want to divide $10 by 0. That means we would want to find a number that we multiply by 0 to get 10. This cannot happen because 0 times any number is 0. Division by zero is said to beundefined

Zero divided by any number is 0 ÷ a = 0 Dividing a number by zero is undefined a ÷ 0 undefined Table 1.61

Another way to explain why division by zero is undefined is to remember that division is really repeated subtraction How many times can we take away 0 from 10? Because subtracting 0 will never change the total, we will never get an answer So we cannot divide a number by 0.

EXAMPLE 1.60

Divide Check by multiplying:ⓐ0 ÷ 3 ⓑ 10 / 0.

Solution

ⓐ

0 ÷ 3

Zero divided by any number is zero 0

Check by multiplying

0 · 3 0 ✓

ⓑ

10 / 0

Division by zero is undefined undefined

TRY IT : :1.119 Divide Then check by multiplying:

ⓐ 0 ÷ 2 ⓑ 17 / 0

TRY IT : :1.120 Divide Then check by multiplying:

ⓐ 0 ÷ 6 ⓑ 13 / 0

Divide the first digit of the dividend, 7, by the divisor,

The divisor can go into two times since 2×3 = 6 Write the above the in the quotient

Multiply the in the quotient by and write the product, 6, under the

Subtract that product from the first digit in the dividend Subtract 7 − 6 Write the difference, 1, under the first digit in the dividend

Bring down the next digit of the dividend Bring down the Divide 18 by the divisor, The divisor goes into 18 six times Write in the quotient above the

Multiply the in the quotient by the divisor and write the product, 18, under the dividend Subtract 18 from 18

We would repeat the process until there are no more digits in the dividend to bring down In this problem, there are no more digits to bring down, so the division is finished

So 78 ÷ = 26.

Check by multiplying the quotient times the divisor to get the dividend Multiply 26 × 3 to make sure that product equals the dividend, 78.

216 ×3 _

78 ✓ It does, so our answer is correct

HOW TO : :DIVIDE WHOLE NUMBERS

Divide the first digit of the dividend by the divisor

If the divisor is larger than the first digit of the dividend, divide the first two digits of the dividend by the divisor, and so on

Write the quotient above the dividend

Multiply the quotient by the divisor and write the product under the dividend Subtract that product from the dividend

Bring down the next digit of the dividend

Repeat from Step until there are no more digits in the dividend to bring down Check by multiplying the quotient times the divisor

Divide 2,596 ÷ 4.Check by multiplying:

Solution

Let's rewrite the problem to set it up for long division Divide the first digit of the dividend, 2, by the divisor,

Since does not go into 2, we use the first two digits of the dividend and divide 25 by The divisor goes into 25 six times

We write the in the quotient above the

Multiply the in the quotient by the divisor and write the product, 24, under the first two digits in the dividend

Subtract that product from the first two digits in the dividend Subtract 25 − 24 Write the difference, 1, under the second digit in the dividend

Now bring down the and repeat these steps There are fours in 19 Write the over the Multiply the by and subtract this product from 19

Bring down the and repeat these steps There are fours in 36 Write the over the Multiply the by and subtract this product from 36

So 2,596 ÷ = 649 Check by multiplying

It equals the dividend, so our answer is correct

TRY IT : :1.121 Divide Then check by multiplying: 2,636 ÷ 4

TRY IT : :1.122 Divide Then check by multiplying: 2,716 ÷ 4 EXAMPLE 1.62

Solution

Let's rewrite the problem to set it up for long division First we try to divide into

Since that won't work, we try into 45

There are sixes in 45 We write the over the Multiply the by and subtract this product from 45

Now bring down the and repeat these steps There are sixes in 30 Write the over the Multiply the by and subtract this product from 30

Now bring down the and repeat these steps There is six in Write the over the Multiply by and subtract this product from

Check by multiplying

It equals the dividend, so our answer is correct

TRY IT : :1.123 Divide Then check by multiplying: 4,305 ÷ 5. TRY IT : :1.124 Divide Then check by multiplying: 3,906 ÷ 6. EXAMPLE 1.63

Let's rewrite the problem to set it up for long division First we try to divide into

Since that won't work, we try into 72 There are nines in 72 We write the over the

Multiply the by and subtract this product from 72

Now bring down the and repeat these steps There are nines in Write the over the Multiply the by and subtract this product from

Now bring down the and repeat these steps There are nines in 63 Write the over the Multiply the by and subtract this product from 63

Check by multiplying

It equals the dividend, so our answer is correct

TRY IT : :1.125 Divide Then check by multiplying: 4,928 ÷ 7. TRY IT : :1.126 Divide Then check by multiplying: 5,663 ÷ 7.

So far all the division problems have worked out evenly For example, if we had 24 cookies and wanted to make bags of 8 cookies, we would have 3 bags But what if there were 28 cookies and we wanted to make bags of 8? Start with the 28 cookies as shown inFigure 1.14

Figure 1.14

Figure 1.15

There are 3 groups of eight cookies, and 4 cookies left over We call the 4 cookies that are left over the remainder and show it by writing R4 next to the 3. (The R stands for remainder.)

To check this division we multiply 3 times 8 to get 24, and then add the remainder of 4. 3

×8 _

24 +4 _ 28 EXAMPLE 1.64

Divide 1,439 ÷ 4.Check by multiplying

Solution

Let's rewrite the problem to set it up for long division

First we try to divide into Since that won't work, we try into 14 There are fours in 14 We write the over the

Multiply the by and subtract this product from 14

Now bring down the and repeat these steps There are fours in 23 Write the over the Multiply the by and subtract this product from 23

Now bring down the and repeat these steps There are fours in 39 Write the over the Multiply the by and subtract this product from 39 There are no more numbers to bring down, so we are done

The remainder is Check by multiplying

TRY IT : :1.128 Divide Then check by multiplying: 4,319 ÷ 8.

EXAMPLE 1.65

Divide and then check by multiplying: 1,461 ÷ 13.

Solution

Let's rewrite the problem to set it up for long division 13 1,461 First we try to divide 13 into Since that won't work, we try 13 into 14

There is thirteen in 14 We write the over the Multiply the by 13 and subtract this product from 14

Now bring down the and repeat these steps There is thirteen in 16 Write the over the Multiply the by 13 and subtract this product from 16

Now bring down the and repeat these steps There are thirteens in 31

Write the over the Multiply the by 13 and subtract this product from 31 There are no more numbers to bring down, so we are done

The remainder is 1,462 ÷ 13 is 112 with a remainder of Check by multiplying

Our answer is correct

TRY IT : :1.129 Divide Then check by multiplying: 1,493 ÷ 13. TRY IT : :1.130 Divide Then check by multiplying: 1,461 ÷ 12. EXAMPLE 1.66

Solution

Let's rewrite the problem to set it up for long division 241 74,521 First we try to divide 241 into Since that won’t work, we try 241 into 74 That still

won’t work, so we try 241 into 745 Since divides into three times, we try Since 3×241 = 723, we write the over the in 745

Note that would be too large because 4×241 = 964, which is greater than 745 Multiply the by 241 and subtract this product from 745

Now bring down the and repeat these steps 241 does not divide into 222 We write a over the as a placeholder and then continue

Now bring down the and repeat these steps Try Since 9×241 = 2,169, we write the over the Multiply the by 241 and subtract this product from 2,221

There are no more numbers to bring down, so we are finished The remainder is 52 So 74,521 ÷ 241

is 309 with a remainder of 52 Check by multiplying

Sometimes it might not be obvious how many times the divisor goes into digits of the dividend We will have to guess and check numbers to find the greatest number that goes into the digits without exceeding them

TRY IT : :1.131 Divide Then check by multiplying: 78,641 ÷ 256. TRY IT : :1.132 Divide Then check by multiplying: 76,461 ÷ 248.

Translate Word Phrases to Math Notation

Operation Word Phrase Example Expression

Division divided by quotient of divided into

12 divided by 4

the quotient of 12 and 4 4 divided into 12

12 ÷ 4 12

4 12/4 4 12

Table 1.70

EXAMPLE 1.67

Translate and simplify: the quotient of 51 and 17.

Solution

The wordquotienttells us to divide the quotient of 51 and 17 Translate. 51 ÷ 17

Divide. 3

We could just as correctly have translatedthe quotient of 51 and 17 using the notation 17 51 or 5117.

TRY IT : :1.133 Translate and simplify: the quotient of 91 and 13. TRY IT : :1.134 Translate and simplify: the quotient of 52 and 13.

Divide Whole Numbers in Applications

We will use the same strategy we used in previous sections to solve applications First, we determine what we are looking for Then we write a phrase that gives the information to find it We then translate the phrase into math notation and simplify it to get the answer Finally, we write a sentence to answer the question

EXAMPLE 1.68

Cecelia bought a 160-ounce box of oatmeal at the big box store She wants to divide the 160 ounces of oatmeal into 8-ounce servings She will put each serving into a plastic bag so she can take one bag to work each day How many servings will she get from the big box?

Solution

We are asked to find the how many servings she will get from the big box

Write a phrase 160 ounces divided by ounces Translate to math notation 160 ÷ 8

Simplify by dividing 20

Write a sentence to answer the question Cecelia will get 20 servings from the big box

TRY IT : :1.135

TRY IT : :1.136

Andrea is making bows for the girls in her dance class to wear at the recital Each bow takes 4 feet of ribbon, and 36 feet of ribbon are on one spool How many bows can Andrea make from one spool of ribbon?

MEDIA : :ACCESS ADDITIONAL ONLINE RESOURCES

• Dividing Whole Numbers (http://www.openstaxcollege.org/l/24divwhlnum)

• Dividing Whole Numbers No Remainder (http://www.openstaxcollege.org/l/24divnumnorem)

Practice Makes Perfect

Use Division Notation

In the following exercises, translate from math notation to words.

343. 54 ÷ 9 344. 56

7 345. 328

346. 6 42 347.48 ÷ 6 348. 63

9

349. 7 63 350.72 ÷ 8

Model Division of Whole Numbers

In the following exercises, model the division.

351. 15 ÷ 5 352. 10 ÷ 5 353. 14

7 354. 18

6 355. 4 20 356. 3 15

357. 24 ÷ 6 358. 16 ÷ 4

Divide Whole Numbers

In the following exercises, divide Then check by multiplying.

359. 18 ÷ 2 360. 14 ÷ 2 361. 27

3

362. 303 363. 4 28 364. 4 36

365. 455 366. 355 367. 72 / 8

368. 8 64 369. 35

7 370. 42 ÷ 7

371. 15 15 372. 12 12 373. 43 ÷ 43

374. 37 ÷ 37 375. 23

1 376. 291

377. 19 ÷ 1 378. 17 ÷ 1 379. 0 ÷ 4

380. 0 ÷ 8 381. 5

0 382. 90

383. 26

0 384. 320 385. 12 0

386. 16 0 387. 72 ÷ 3 388. 57 ÷ 3

389. 96

8 390. 786 391. 5 465

392. 4 528 393. 924 ÷ 7 394. 861 ÷ 7

395. 5,2266 396. 3,7768 397. 4 31,324

398. 5 46,855 399.7,209 ÷ 3 400. 4,806 ÷ 3

401. 5,406 ÷ 6 402. 3,208 ÷ 4 403. 4 2,816

404. 6 3,624 405. 91,881

9 406. 83,2568

407. 2,470 ÷ 7 408. 3,741 ÷ 7 409. 8 55,305

410. 9 51,492 411. 431,174

5 412. 297,2774

413. 130,016 ÷ 3 414.105,609 ÷ 2 415. 15 5,735

416. 4,93321 417. 56,883 ÷ 67 418. 43,725 / 75

419. 30,144314 420. 26,145 ÷ 415 421. 273 542,195

422. 816,243 ÷ 462

Mixed Practice

In the following exercises, simplify.

423. 15(204) 424. 74 · 391 425. 256 − 184

426. 305 − 262 427.719 + 341 428. 647 + 528

429. 25 875 430. 1104 ÷ 23

Translate Word Phrases to Algebraic Expressions

In the following exercises, translate and simplify.

In the following exercises, solve.

435. Trail mix Ric bought 64 ounces of trail mix He wants to divide it into small bags, with 2 ounces of trail mix in each bag How many bags can Ric fill?

436.CrackersEvie bought a 42 ounce box of crackers She wants to divide it into bags with 3 ounces of crackers in each bag How many bags can Evie fill?

437. Astronomy class There are 125 students in an astronomy class The professor assigns them into groups of 5. How many groups of students are there? 438.Flower shopMelissa’s flower

shop got a shipment of 152 roses She wants to make bouquets of 8 roses each How many bouquets can Melissa make?

439. Baking One roll of plastic wrap is 48 feet long Marta uses 3 feet of plastic wrap to wrap each cake she bakes How many cakes can she wrap from one roll?

440.Dental flossOne package of dental floss is 54 feet long Brian uses 2 feet of dental floss every day How many days will one package of dental floss last Brian?

Mixed Practice

In the following exercises, solve.

441. Miles per gallon Susana’s hybrid car gets 45 miles per gallon Her son’s truck gets 17 miles per gallon What is the difference in miles per gallon between Susana’s car and her son’s truck?

442. Distance Mayra lives 53 miles from her mother’s house and 71 miles from her mother-in-law’s house How much farther is Mayra from her mother-in-law’s house than from her mother’s house?

443.Field tripThe 45 students in a Geology class will go on a field trip, using the college’s vans Each van can hold 9 students How many vans will they need for the field trip?

444. Potting soil Aki bought a 128 ounce bag of potting soil How many 4 ounce pots can he fill from the bag?

445.HikingBill hiked 8 miles on the first day of his backpacking trip, 14 miles the second day, 11 miles the third day, and 17 miles the fourth day What is the total number of miles Bill hiked?

446. Reading Last night Emily read 6 pages in her Business textbook, 26 pages in her History text, 15 pages in her Psychology text, and 9 pages in her math text What is the total number of pages Emily read?

447.PatientsLaVonne treats 12 patients each day in her dental office Last week she worked 4 days How many patients did she treat last week?

448.ScoutsThere are 14 boys in Dave’s scout troop At summer camp, each boy earned 5 merit badges What was the total number of merit badges earned by Dave’s scout troop at summer camp?

Writing Exercises

449. Explain how you use the multiplication facts to

help with division 450.wasOswaldo divided37 with a remainder of300 by4.8How can you check toand said his answer make sure he is correct?

Everyday Math

451.Contact lensesJenna puts in a new pair of contact lenses every 14 days How many pairs of contact lenses does she need for 365 days?

Self Check

ⓐAfter completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

coordinate counting numbers difference dividend divisor number line origin

place value system product quotient rounding sum whole numbers KEY TERMS

A number paired with a point on a number line is called the coordinate of the point The counting numbers are the numbers 1, 2, 3, …

The difference is the result of subtracting two or more numbers When dividing two numbers, the dividend is the number being divided When dividing two numbers, the divisor is the number dividing the dividend

A number line is used to visualize numbers The numbers on the number line get larger as they go from left to right, and smaller as they go from right to left

The origin is the point labeled on a number line

Our number system is called a place value system because the value of a digit depends on its position, or place, in a number

The product is the result of multiplying two or more numbers The quotient is the result of dividing two numbers

The process of approximating a number is called rounding The sum is the result of adding two or more numbers

The whole numbers are the numbers 0, 1, 2, 3, …

KEY CONCEPTS

1.1 Introduction to Whole Numbers

Figure 1.16

• Name a whole number in words.

Starting at the digit on the left, name the number in each period, followed by the period name Do not include the period name for the ones

Use commas in the number to separate the periods • Use place value to write a whole number.

Identify the words that indicate periods (Remember the ones period is never named.) Draw three blanks to indicate the number of places needed in each period

Name the number in each period and place the digits in the correct place value position • Round a whole number to a specific place value.

Locate the given place value All digits to the left of that place value not change Underline the digit to the right of the given place value

Determine if this digit is greater than or equal to If yes—add to the digit in the given place value If no—do not change the digit in the given place value

Replace all digits to the right of the given place value with zeros

1.2 Add Whole Numbers

• Addition NotationTo describe addition, we can use symbols and words

Operation Notation Expression Read as Result

Addition + 3 + 4 three plus four the sum of 3 and 4 • Identity Property of Addition

◦ The sum of any number a and 0 is the number a + = a + a = a • Commutative Property of Addition

◦ Changing the order of the addends a and b does not change their sum a + b = b + a • Add whole numbers.

Write the numbers so each place value lines up vertically

Add the digits in each place value Work from right to left starting with the ones place If a sum in a place value is more than 9, carry to the next place value

Continue adding each place value from right to left, adding each place value and carrying if needed

1.3 Subtract Whole Numbers

Operation Notation Expression Read as Result

Subtraction − 7 − 3 seven minus three the difference of 7 and 3 • Subtract whole numbers.

Write the numbers so each place value lines up vertically

Subtract the digits in each place value Work from right to left starting with the ones place If the digit on top is less than the digit below, borrow as needed

Continue subtracting each place value from right to left, borrowing if needed Check by adding

1.4 Multiply Whole Numbers

Operation Notation Expression Read as Result

Multiplication × · ( )

3 × 8 3 · 8 3(8)

three times eight the product of and 8

• Multiplication Property of Zero

◦ The product of any number and is

a ⋅ = 0

0 ⋅ a = 0

• Identity Property of Multiplication

◦ The product of any number and is the number 1 ⋅ a = a

a ⋅ = a

• Commutative Property of Multiplication

• Multiply two whole numbers to find the product.

Write the numbers so each place value lines up vertically Multiply the digits in each place value

Work from right to left, starting with the ones place in the bottom number

Multiply the bottom number by the ones digit in the top number, then by the tens digit, and so on If a product in a place value is more than 9, carry to the next place value

Write the partial products, lining up the digits in the place values with the numbers above Repeat for the tens place in the bottom number, the hundreds place, and so on

Insert a zero as a placeholder with each additional partial product Add the partial products

1.5 Divide Whole Numbers

Operation Notation Expression Read as Result

Division ÷

a b b a a/ b

12 ÷ 4 12

4 4 12 12 / 4

Twelve divided by four the quotient of 12 and 4

• Division Properties of One

◦ Any number (except 0) divided by itself is one a ÷ a = 1 ◦ Any number divided by one is the same number a ÷ = a

• Division Properties of Zero

◦ Zero divided by any number is 0 ÷ a = 0

◦ Dividing a number by zero is undefined a ữ 0 undefined ã Divide whole numbers.

Divide the first digit of the dividend by the divisor

If the divisor is larger than the first digit of the dividend, divide the first two digits of the dividend by the divisor, and so on

Write the quotient above the dividend

Multiply the quotient by the divisor and write the product under the dividend Subtract that product from the dividend

Bring down the next digit of the dividend

Repeat from Step until there are no more digits in the dividend to bring down Check by multiplying the quotient times the divisor

REVIEW EXERCISES

1.1 Introduction to Whole Numbers

Identify Counting Numbers and Whole Numbers

In the following exercises, determine which of the following are (a) counting numbers (b) whole numbers.

453. 0, 2, 99 454. 0, 3, 25 455. 0, 4, 90

Model Whole Numbers

In the following exercises, model each number using base-10blocks and then show its value using place value notation.

457. 258 458. 104

Identify the Place Value of a Digit

In the following exercises, find the place value of the given digits.

459. 472,981

ⓐ 8 ⓑ 4 ⓒ 1

ⓓ 7 ⓔ 2

460. 12,403,295

ⓐ 4 ⓑ 0 ⓒ 1

ⓓ 9 ⓔ 3

Use Place Value to Name Whole Numbers

In the following exercises, name each number in words.

461. 5,280 462. 204,614 463. 5,012,582

464. 31,640,976

Use Place Value to Write Whole Numbers

In the following exercises, write as a whole number using digits.

465. six hundred two 466. fifteen thousand, two

hundred fifty-three 467.nine hundred twelve thousand,three hundred forty million, sixty-one

468. two billion, four hundred ninety-two million, seven hundred eleven thousand, two

Round Whole Numbers

In the following exercises, round to the nearest ten.

469. 412 470. 648 471. 3,556

472. 2,734

In the following exercises, round to the nearest hundred.

473. 38,975 474. 26,849 475. 81,486

476. 75,992

1.2 Add Whole Numbers

Use Addition Notation

In the following exercises, translate the following from math notation to words.

477. 4 + 3 478. 25 + 18 479. 571 + 629

480. 10,085 + 3,492

Model Addition of Whole Numbers

In the following exercises, model the addition.

In the following exercises, fill in the missing values in each chart.

483. 484.

In the following exercises, add.

485. ⓐ 0 + 19 ⓑ 19 + 0 486. ⓐ 0 + 480 ⓑ 480 + 0 487. ⓐ 7 + 6 ⓑ 6 + 7

488. ⓐ 23 + 18 ⓑ 18 + 23 489. 44 + 35 490. 63 + 29

491. 96 + 58 492. 375 + 591 493. 7,281 + 12,546

494. 5,280 + 16,324 + 9,731

Translate Word Phrases to Math Notation

In the following exercises, translate each phrase into math notation and then simplify.

495. the sum of 30 and 12 496. 11 increased by 8 497. 25 more than 39 498. total of 15 and 50

Add Whole Numbers in Applications

In the following exercises, solve.

499. Shopping for an interview

Nathan bought a new shirt, tie, and slacks to wear to a job interview The shirt cost $24, the tie cost $14, and the slacks cost $38. What was Nathan’s total cost?

500. RunningJackson ran 4miles on Monday, 12 miles on Tuesday, 1 mile on Wednesday, 8miles on Thursday, and 5 miles on Friday What was the total number of miles Jackson ran?

In the following exercises, find the perimeter of each figure.

501. 502.

1.3 Subtract Whole Numbers

Use Subtraction Notation

506. 5,724 − 2,918

Model Subtraction of Whole Numbers

In the following exercises, model the subtraction.

507. 18 − 4 508. 41 − 29

Subtract Whole Numbers

In the following exercises, subtract and then check by adding.

509. 8 − 5 510. 12 − 7 511. 23 − 9

512. 46 − 21 513. 82 − 59 514. 110 − 87

515. 539 − 217 516. 415 − 296 517. 1,020 − 640

518. 8,355 − 3,947 519. 10,000 − 15 520. 54,925 − 35,647

Translate Word Phrases to Math Notation

In the following exercises, translate and simplify.

521. the difference of nineteen

and thirteen 522.hundredsubtract sixty-five from one 523.eight seventy-four decreased by 524. twenty-three less than

forty-one

Subtract Whole Numbers in Applications

In the following exercises, solve.

525. Temperature The high temperature in Peoria one day was 86 degrees Fahrenheit and the low temperature was 28 degrees Fahrenheit What was the difference between the high and low temperatures?

526. SavingsLynn wants to go on a cruise that costs $2,485. She has $948 in her vacation savings account How much more does she need to save in order to pay for the cruise?

1.4 Multiply Whole Numbers

Use Multiplication Notation

In the following exercises, translate from math notation to words.

527. 8 × 5 528. 6 · 14 529. (10)(95)

530. 54(72)

Model Multiplication of Whole Numbers

In the following exercises, model the multiplication.

In the following exercises, fill in the missing values in each chart.

533. 534.

In the following exercises, multiply.

535. 0 · 14 536. (256)0 537. 1 · 99

538. (4,789)1 539. ⓐ 7 · 4 ⓑ 4 · 7 540. (25)(6)

541. 9,261 × 3 542. 48 · 76 543. 64 · 10

544. 1,000(22) 545. 162 × 493 546. (601)(943)

547. 3,624 × 517 548. 10,538 · 22

Translate Word Phrases to Math Notation

In the following exercises, translate and simplify.

549. the product of 15 and 28 550. ninety-four times

thirty-three 551. twice 575

552. ten times two hundred sixty-four

Multiply Whole Numbers in Applications

In the following exercises, solve.

553. Gardening Geniece bought 8 packs of marigolds to plant in her yard Each pack has 6 flowers How many marigolds did Geniece buy?

554. CookingRatika is making rice for a dinner party The number of cups of water is twice the number of cups of rice If Ratika plans to use 4cups of rice, how many cups of water does she need?

555. Multiplex There are twelve theaters at the multiplex and each theater has 150 seats What is the total number of seats at the multiplex?

556. RoofingLewis needs to put new shingles on his roof The roof is a rectangle, 30 feet by 24 feet What is the area of the roof?

1.5 Divide Whole Numbers

Use Division Notation

Translate from math notation to words.

557. 54 ÷ 9 558. 42 / 7 559. 72

560. 6 48

Model Division of Whole Numbers

In the following exercises, model.

561. 8 ÷ 2 562. 3 12

Divide Whole Numbers

In the following exercises, divide Then check by multiplying.

563. 14 ÷ 2 564. 32

8 565. 52 ÷ 4

566. 26 26 567. 97

1 568. 0 ÷ 52

569. 100 ÷ 0 570. 355

5 571. 3828 ÷ 6

572. 31 1,519 573. 7505

25 574. 5,166 ÷ 42

Translate Word Phrases to Math Notation

In the following exercises, translate and simplify.

575. the quotient of 64 and 16 576. the quotient of 572and 52

Divide Whole Numbers in Applications

In the following exercises, solve.

577. RibbonOne spool of ribbon is 27 feet Lizbeth uses 3 feet of ribbon for each gift basket that she wraps How many gift baskets can Lizbeth wrap from one spool of ribbon?

579. Determine which of the following numbers are

ⓐcounting numbers

ⓑwhole numbers 0, 4, 87

580. Find the place value of the given digits in the number

549,362.

ⓐ 9 ⓑ 6 ⓒ 2

ⓓ 5

581. Write each number as a whole number using digits

ⓐsix hundred thirteen

ⓑ fifty-five thousand two hundred eight 582. Round 25,849 to the nearest

hundred

Simplify.

583. 45 + 23 584. 65 − 42 585. 85 ữ 5

586. 1,000 ì 8 587. 90 − 58 588. 73 + 89

589. (0)(12,675) 590. 634 + 255 591. 0

9

592. 8 128 593. 145 − 79 594. 299 + 836

595. 7 · 475 596. 8,528 + 704 597. 35(14)

598. 260 599. 733 − 291 600. 4,916 − 1,538

601. 495 ÷ 45 602. 52 × 983

Translate each phrase to math notation and then simplify.

603. The sum of 16 and 58 604. The product of 9 and 15 605. The difference of 32 and 18

606. The quotient of 63 and 21 607. Twice 524 608. 29 more than 32 609. 50 less than 300

In the following exercises, solve.

610. LaVelle buys a jumbo bag of 84 candies to make favor bags for her son’s party If she wants to make 12 bags, how many candies should she put in each bag?

611. Last month, Stan’s take-home pay was $3,816 and his expenses were $3,472. How much of his take-home pay did Stan have left after he paid his expenses?

612. Each class at Greenville School has 22 children enrolled The school has 24 classes How many children are enrolled at Greenville School?

Figure 2.1 Algebra has a language of its own The picture shows just some of the words you may see and use in your study of Prealgebra

Chapter Outline

2.1Use the Language of Algebra

2.2Evaluate, Simplify, and Translate Expressions

2.3Solving Equations Using the Subtraction and Addition Properties of Equality

2.4Find Multiples and Factors

2.5Prime Factorization and the Least Common Multiple Introduction

You may not realize it, but you already use algebra every day Perhaps you figure out how much to tip a server in a restaurant Maybe you calculate the amount of change you should get when you pay for something It could even be when you compare batting averages of your favorite players You can describe the algebra you use in specific words, and follow an orderly process In this chapter, you will explore the words used to describe algebra and start on your path to solving algebraic problems easily, both in class and in your everyday life

2.1 Use the Language of Algebra Learning Objectives

By the end of this section, you will be able to:

Use variables and algebraic symbols Identify expressions and equations Simplify expressions with exponents

Simplify expressions using the order of operations

Be Prepared!

Before you get started, take this readiness quiz Add: 43 + 69.

If you missed this problem, reviewExample 1.19 Multiply: (896)201.

If you missed this problem, reviewExample 1.48 Divide: 7,263 ÷ 9.

Greg and Alex have the same birthday, but they were born in different years This year Greg is 20 years old and Alex is 23, so Alex is 3 years older than Greg When Greg was 12, Alex was 15. When Greg is 35, Alex will be 38. No matter what Greg’s age is, Alex’s age will always be 3 years more, right?

In the language of algebra, we say that Greg’s age and Alex’s age are variable and the three is a constant The ages change, or vary, so age is a variable The 3 years between them always stays the same, so the age difference is the constant

In algebra, letters of the alphabet are used to represent variables Suppose we call Greg’s age g. Then we could use

g + 3 to represent Alex’s age SeeTable 2.1

Greg’s age Alex’s age

12 15

20 23

35 38

g g + 3

Table 2.1

Letters are used to represent variables Letters often used for variables are x, y, a, b, and c.

Variables and Constants

A variable is a letter that represents a number or quantity whose value may change A constant is a number whose value always stays the same

To write algebraically, we need some symbols as well as numbers and variables There are several types of symbols we will be using InWhole Numbers, we introduced the symbols for the four basic arithmetic operations: addition, subtraction, multiplication, and division We will summarize them here, along with words we use for the operations and the result

Operation Notation Say: The result is…

Addition a + b a plus b the sum of a and b Subtraction a − b a minus b the difference of a and b Multiplication a · b, (a)(b), (a)b, a(b) a times b The product of a and b Division a ÷ b, a / b, ab, b a a divided by b The quotient of a and b

In algebra, the cross symbol, ×, is not used to show multiplication because that symbol may cause confusion Does 3xy mean 3 × y (three times y) or 3 · x · y (three times x times y)? To make it clear, use • or parentheses for multiplication We perform these operations on two numbers When translating from symbolic form to words, or from words to symbolic form, pay attention to the wordsoforandto help you find the numbers

Thesumof 5and 3means add 5 plus 3, which we write as 5 + 3.

EXAMPLE 2.1

Translate from algebra to words:

ⓐ 12 + 14 ⓑ (30)(5) ⓒ 64 ÷ 8 ⓓ x − y

Solution

ⓐ

12 + 14

12 plus 14

the sum of twelve and fourteen

ⓑ

(30)(5)

30 times

the product of thirty and five

ⓒ

64 ÷ 8

64 divided by

the quotient of sixty-four and eight

ⓓ

x − y x minus y the difference of x and y

TRY IT : :2.1 Translate from algebra to words.

ⓐ18 + 11 ⓑ(27)(9) ⓒ 84 ÷ 7 ⓓp − q

TRY IT : :2.2 Translate from algebra to words

ⓐ 47 − 19 ⓑ 72 ÷ 9 ⓒ m + n ⓓ(13)(7)

When two quantities have the same value, we say they are equal and connect them with anequal sign

Equality Symbol

An inequality is used in algebra to compare two quantities that may have different values The number line can help you understand inequalities Remember that on the number line the numbers get larger as they go from left to right So if we know that b is greater than a, it means that b is to the right of a on the number line We use the symbols “<” and

“>” for inequalities

Inequality

a < b is read a is less than b

a is to the left of b on the number line

a > b is read a is greater than b a is to the right of b on the number line

The expressions a < b and a > b can be read from left-to-right or right-to-left, though in English we usually read from left-to-right In general,

a < b is equivalent to b > a. For example, < 11 is equivalent to 11 > 7.

a > b is equivalent to b < a. For example, 17 > is equivalent to < 17.

When we write an inequality symbol with a line under it, such as a ≤ b, it means a < b or a = b. We read this a is less than or equal to b. Also, if we put a slash through an equal sign, ≠, it means not equal

We summarize the symbols of equality and inequality inTable 2.6

Algebraic Notation Say

a = b a is equal to b a ≠ b a is not equal to b a < b a is less than b a > b a is greater than b a ≤ b a is less than or equal to b a ≥ b a is greater than or equal to b

Table 2.6 Symbols < and >

The symbols < and > each have a smaller side and a larger side smaller side < larger side

larger side > smaller side

EXAMPLE 2.2

Translate from algebra to words:

ⓐ 20 ≤ 35 ⓑ 11 ≠ 15 − 3 ⓒ 9 > 10 ÷ 2 ⓓ x + < 10

Solution

ⓐ

20 ≤ 35

20 is less than or equal to 35

ⓑ

11 ≠ 15 − 3

11 is not equal to 15 minus

ⓒ

9 > 10 ÷ 2

9 is greater than 10 divided by

ⓓ

x + < 10

x plus is less than 10

TRY IT : :2.3 Translate from algebra to words.

ⓐ 14 ≤ 27 ⓑ 19 − ≠ 8 ⓒ12 > ÷ 2 ⓓ x − < 1

TRY IT : :2.4 Translate from algebra to words.

The information inFigure 2.2compares the fuel economy in miles-per-gallon (mpg) of several cars Write the appropriate symbol =, <, or > in each expression to compare the fuel economy of the cars

Figure 2.2 (credit: modification of work by Bernard Goldbach, Wikimedia Commons)

ⓐMPG of Prius _ MPG of Mini Cooper ⓑMPG of Versa _ MPG of Fit

ⓒMPG of Mini Cooper _ MPG of Fit ⓓMPG of Corolla _ MPG of Versa

ⓔMPG of Corolla _ MPG of Prius

Solution

ⓐ

MPG of Prius MPG of Mini Cooper Find the values in the chart 48 27

Compare 48 > 27

MPG of Prius > MPG of Mini Cooper

ⓑ

MPG of Versa MPG of Fit Find the values in the chart 26 27

Compare 26 < 27

MPG of Versa < MPG of Fit

ⓒ

MPG of Mini Cooper MPG of Fit Find the values in the chart 27 27

Compare 27 = 27

ⓓ

MPG of Corolla MPG of Versa Find the values in the chart 28 26

Compare 28 > 26

MPG of Corolla > MPG of Versa

ⓔ

MPG of Corolla MPG of Prius Find the values in the chart 28 48

Compare 28 < 48

MPG of Corolla < MPG of Prius

TRY IT : :2.5 UseFigure 2.2to fill in the appropriate symbol, =, <, or >.

ⓐMPG of Prius _MPG of Versa ⓑMPG of Mini Cooper _ MPG of Corolla TRY IT : :2.6 UseFigure 2.2to fill in the appropriate symbol, =, <, or >.

ⓐMPG of Fit _ MPG of Prius ⓑMPG of Corolla _ MPG of Fit

Grouping symbols in algebra are much like the commas, colons, and other punctuation marks in written language They indicate which expressions are to be kept together and separate from other expressions.Table 2.16lists three of the most commonly used grouping symbols in algebra

Common Grouping Symbols

parentheses ( ) brackets [ ]

braces { }

Table 2.16

Here are some examples of expressions that include grouping symbols We will simplify expressions like these later in this section

8(14 − 8) 21 − 3[2 + 4(9 − 8)] 24 ÷⎧ ⎩

⎨13 − 2[1(6 − 5) + 4]⎫ ⎭ ⎬

Identify Expressions and Equations

What is the difference in English between a phrase and a sentence? A phrase expresses a single thought that is incomplete by itself, but a sentence makes a complete statement “Running very fast” is a phrase, but “The football player was running very fast” is a sentence A sentence has a subject and a verb

Expression Words Phrase

3 + 5 3 plus 5 the sum of three and five n − 1 n minus one the difference of n and one

6 · 7 6 times 7 the product of six and seven x

y x divided by y the quotient of x and y

Notice that the phrases not form a complete sentence because the phrase does not have a verb An equation is two expressions linked with an equal sign When you read the words the symbols represent in an equation, you have a complete sentence in English The equal sign gives the verb Here are some examples of equations:

Equation Sentence

3 + = 8 The sum of three and five is equal to eight n − = 14 n minus one equals fourteen

6 · = 42 The product of six and seven is equal to forty-two x = 53 x is equal to fifty-three

y + = 2y − 3 y plus nine is equal to two y minus three Expressions and Equations

Anexpressionis a number, a variable, or a combination of numbers and variables and operation symbols Anequationis made up of two expressions connected by an equal sign

EXAMPLE 2.4

Determine if each is an expression or an equation:

ⓐ 16 − = 10 ⓑ 4 · + 1 ⓒ x ÷ 25 ⓓ y + = 40

Solution

ⓐ 16 − = 10 This is an equation—two expressions are connected with an equal sign

ⓑ 4 · + 1 This is an expression—no equal sign

ⓒ x ÷ 25 This is an expression—no equal sign

ⓓ y + = 40 This is an equation—two expressions are connected with an equal sign

TRY IT : :2.7 Determine if each is an expression or an equation:

TRY IT : :2.8 Determine if each is an expression or an equation:

y ÷ 14 x − = 21

Simplify Expressions with Exponents

To simplify a numerical expression means to all the math possible For example, to simplify 4 · + 1 we’d first multiply 4 · 2 to get 8 and then add the 1 to get 9. A good habit to develop is to work down the page, writing each step of the process below the previous step The example just described would look like this:

4 · + 1 8 + 1

9

Suppose we have the expression 2 · · · · · · · · 2. We could write this more compactly using exponential notation Exponential notation is used in algebra to represent a quantity multiplied by itself several times We write 2 · · 2 as 23 and 2 · · · · · · · · 2 as 29. In expressions such as 23, the 2 is called thebaseand the 3 is called the exponent The exponent tells us how many factors of the base we have to multiply

means multiply three factors of 2 We say 23 is in exponential notation and 2 · · 2 is in expanded notation

Exponential Notation

For any expression an, a is a factor multiplied by itself n times if n is a positive integer

anmeans multiply n factors of a

The expression an is read a to the nth power

For powers of n = 2 and n = 3, we have special names

a2is read as "a squared"

a3is read as "a cubed"

Table 2.18lists some examples of expressions written in exponential notation

Exponential Notation In Words

72 7 to the second power, or 7 squared

53 5 to the third power, or 5 cubed

94 9 to the fourth power

Write each expression in exponential form:

ⓐ 16 · 16 · 16 · 16 · 16 · 16 · 16 ⓑ9 · · · · 9 ⓒ x · x · x · x ⓓ a · a · a · a · a · a · a · a

Solution

ⓐThe base 16 is a factor times 167

ⓑThe base is a factor times 95

ⓒThe base x is a factor times x4

ⓓThe base a is a factor times a8

TRY IT : :2.9 Write each expression in exponential form:

41 · 41 · 41 · 41 · 41

TRY IT : :2.10 Write each expression in exponential form: 7 · · · · · · · · 7

EXAMPLE 2.6

Write each exponential expression in expanded form:

ⓐ 86 ⓑ x5

Solution

ⓐThe base is 8and the exponent is 6, so 86 means 8 · · · · · 8

ⓑThe base is x and the exponent is 5, so x5 means x · x · x · x · x

TRY IT : :2.11 Write each exponential expression in expanded form:

ⓐ48 ⓑ a7

TRY IT : :2.12 Write each exponential expression in expanded form:

ⓐ 88 ⓑ b6

To simplify an exponential expression without using a calculator, we write it in expanded form and then multiply the factors

Solution

34

Expand the expression 3 ⋅ ⋅ ⋅ 3 Multiply left to right 9 ⋅ ⋅ 3

27 ⋅ 3

Multiply 81

TRY IT : :2.13 Simplify:

ⓐ 53 ⓑ 17 TRY IT : :2.14 Simplify:

ⓐ72 ⓑ05

Simplify Expressions Using the Order of Operations

We’ve introduced most of the symbols and notation used in algebra, but now we need to clarify the order of operations Otherwise, expressions may have different meanings, and they may result in different values

For example, consider the expression:

4 + · 7

Some students say it simplifies o 49. Some students say it simplifies o 25. 4 + · 7

Since + gives 7. 7 · 7

And · is 49. 49

4 + · 7 Since · is 21. 4 + 21 And 21 + makes 25. 25

Imagine the confusion that could result if every problem had several different correct answers The same expression should give the same result So mathematicians established some guidelines called the order of operations, which outlines the order in which parts of an expression must be simplified

Order of Operations

When simplifying mathematical expressions perform the operations in the following order: 1.Parentheses and other Grouping Symbols

• Simplify all expressions inside the parentheses or other grouping symbols, working on the innermost parentheses first

2.Exponents

• Simplify all expressions with exponents 3.Multiplication andDivision

• Perform all multiplication and division in order from left to right These operations have equal priority 4.Addition andSubtraction

• Perform all addition and subtraction in order from left to right These operations have equal priority

Order of Operations Please Parentheses

Excuse Exponents

MyDear Multiplication andDivision

AuntSally Addition andSubtraction

It’s good that ‘MyDear’ goes together, as this reminds us thatmultiplication anddivision have equal priority We not always multiplication before division or always division before multiplication We them in order from left to right Similarly, ‘AuntSally’ goes together and so reminds us thataddition andsubtraction also have equal priority and we them in order from left to right

MANIPULATIVE MATHEMATICS

Doing the Manipulative Mathematics activity Game of 24 will give you practice using the order of operations

EXAMPLE 2.8

Simplify the expressions:

ⓐ 4 + · 7 ⓑ (4 + 3) · 7

Solution

ⓐ

Are there anyparentheses? No Are there anyexponents? No

Is there anymultiplication ordivision? Yes Multiply first

Add

ⓑ

Are there anyparentheses? Yes Simplify inside the parentheses Are there anyexponents? No

TRY IT : :2.15 Simplify the expressions:

ⓐ 12 − · 2 ⓑ (12 − 5) · 2 TRY IT : :2.16 Simplify the expressions:

ⓐ 8 + · 9 ⓑ(8 + 3) · 9 EXAMPLE 2.9

Simplify:

ⓐ 18 ÷ · 2 ⓑ18 · ÷ 2

Solution

ⓐ

Are there anyparentheses? No Are there anyexponents? No

Is there anymultiplication ordivision? Yes Multiply and divide from left to right Divide Multiply

ⓑ

Are there anyparentheses? No Are there anyexponents? No

Is there anymultiplication ordivision? Yes Multiply and divide from left to right Multiply

Divide

TRY IT : :2.17 Simplify: 42 ÷ · 3 TRY IT : :2.18 Simplify:

12 · ÷ 4 EXAMPLE 2.10

Parentheses? Yes, subtract first Exponents? No

Multiplication or division? Yes

Divide first because we multiply and divide left to right Any other multiplication or division? Yes

Multiply

Any other multiplication or division? No Any addition or subtraction? Yes

TRY IT : :2.19 Simplify:

30 ÷ + 10(3 − 2) TRY IT : :2.20 Simplify:

70 ÷ 10 + 4(6 − 2)

When there are multiple grouping symbols, we simplify the innermost parentheses first and work outward EXAMPLE 2.11

Simplify: + 23+ 3⎡

Solution

Are there any parentheses (or other grouping symbol)? Yes Focus on the parentheses that are inside the brackets Subtract

Continue inside the brackets and multiply Continue inside the brackets and subtract

The expression inside the brackets requires no further simplification Are there any exponents? Yes

Simplify exponents

Is there any multiplication or division? Yes Multiply

Is there any addition or subtraction? Yes Add

Add

TRY IT : :2.21 Simplify: 9 + 53−⎡

⎣4(9 + 3)⎤⎦

TRY IT : :2.22 Simplify:

72− 2⎡

⎣4(5 + 1)⎤⎦

EXAMPLE 2.12

Simplify: 23+ 34÷ − 52.

Solution

If an expression has several exponents, they may be simplified in the same step Simplify exponents

32+ 24÷ + 43 TRY IT : :2.24 Simplify:

62− 53÷ + 82

MEDIA : :ACCESS ADDITIONAL ONLINE RESOURCES

• Order of Operations (http://openstaxcollege.org/l/24orderoperate)

• Order of Operations – The Basics (http://openstaxcollege.org/l/24orderbasic)

• Ex: Evaluate an Expression Using the Order of Operations (http://openstaxcollege.org/l/ 24Evalexpress)

Practice Makes Perfect

Use Variables and Algebraic Symbols

In the following exercises, translate from algebraic notation to words.

1. 16 − 9 2. 25 − 7 3. 5 · 6

4. 3 · 9 5.28 ÷ 4 6. 45 ÷ 5

7. x + 8 8. x + 11 9. (2)(7)

10. (4)(8) 11. 14 < 21 12. 17 < 35

13. 36 ≥ 19 14.42 ≥ 27 15. 3n = 24

16. 6n = 36 17. y − > 6 18. y − > 8

19. 2 ≤ 18 ÷ 6 20. 3 ≤ 20 ÷ 4 21. a ≠ · 4

22. a ≠ · 12

Identify Expressions and Equations

In the following exercises, determine if each is an expression or an equation.

23. 9 · = 54 24. 7 · = 63 25. 5 · + 3

26. 6 · + 5 27. x + 7 28. x + 9

29. y − = 25 30. y − = 32

Simplify Expressions with Exponents

In the following exercises, write in exponential form.

31. 3 · · · · · · 3 32.4 · · · · · 4 33. x · x · x · x · x 34. y · y · y · y · y · y

In the following exercises, write in expanded form.

35. 53 36.83 37. 28

38. 105

Simplify Expressions Using the Order of Operations

In the following exercises, simplify.

39.

ⓐ 3 + · 5

ⓑ (3+8) · 5

40.

ⓐ 2 + · 3

ⓑ (2+6) · 3

41. 23− 12 ÷ (9 − 5)

45. 2 + 8(6 + 1) 46. 4 + 6(3 + 6) 47. 4 · 12 / 8

48. 2 · 36 / 6 49. 6 + 10 / + 2 50. 9 + 12 / + 4

51. (6 + 10) ÷ (2 + 2) 52.(9 + 12) ÷ (3 + 4) 53. 20 ÷ + · 5 54. 33 ÷ + · 2 55. 20 ÷ (4 + 6) · 5 56. 33 ÷ (3 + 8) · 2

57. 42+ 52 58. 32+ 72 59. (4 + 5)2

Everyday Math

65. Basketball In the 2014 NBA playoffs, the San Antonio Spurs beat the Miami Heat The table below shows the heights of the starters on each team Use this table to fill in the appropriate symbol (=, <, >).

Spurs Height Heat Height

Tim

Duncan 83″ RashardLewis 82″ Boris

Diaw 80″ LeBronJames 80″ Kawhi

Leonard 79″ ChrisBosh 83″ Tony

Parker 74″ DwyaneWade 76″ Danny

Green 78″ RayAllen 77″ ⓐ Height of Tim Duncan Height of Rashard Lewis

ⓑHeight of Boris Diaw Height of LeBron James

ⓒ Height of Kawhi Leonard Height of Chris Bosh

ⓓ Height of Tony Parker Height of Dwyane Wade

ⓔHeight of Danny Green Height of Ray Allen

66. Elevation In Colorado there are more than 50 mountains with an elevation of over 14,000 feet. The table shows the ten tallest Use this table to fill in the appropriate inequality symbol

Mountain Elevation

Mt Elbert 14,433′ Mt Massive 14,421′ Mt Harvard 14,420′ Blanca Peak 14,345′ La Plata Peak 14,336′ Uncompahgre Peak 14,309′ Crestone Peak 14,294′ Mt Lincoln 14,286′ Grays Peak 14,270′ Mt Antero 14,269′ ⓐElevation of La Plata Peak Elevation of Mt Antero

ⓑ Elevation of Blanca Peak Elevation of Mt Elbert

ⓒ Elevation of Gray’s Peak Elevation of Mt Lincoln

ⓓ Elevation of Mt Massive Elevation of Crestone Peak

ⓔ Elevation of Mt Harvard Elevation of Uncompahgre Peak

Writing Exercises

67.Explain the difference between an expression and

ⓐAfter completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

ⓑIf most of your checks were:

…confidently Congratulations! You have achieved the objectives in this section Reflect on the study skills you used so that you can continue to use them What did you to become confident of your ability to these things? Be specific.

…with some help This must be addressed quickly because topics you not master become potholes in your road to success. In math, every topic builds upon previous work It is important to make sure you have a strong foundation before you move on. Who can you ask for help? Your fellow classmates and instructor are good resources Is there a place on campus where math tutors are available? Can your study skills be improved?

2.2 Evaluate, Simplify, and Translate Expressions Learning Objectives

By the end of this section, you will be able to:

Evaluate algebraic expressions

Identify terms, coefficients, and like terms Simplify expressions by combining like terms Translate word phrases to algebraic expressions

Be Prepared!

Before you get started, take this readiness quiz Is n ÷ 5 an expression or an equation?

If you missed this problem, reviewExample 2.4 Simplify 45.

If you missed this problem, reviewExample 2.7 Simplify 1 + ⋅ 9.

If you missed this problem, reviewExample 2.8

Evaluate Algebraic Expressions

In the last section, we simplified expressions using the order of operations In this section, we’ll evaluate expressions—again following the order of operations

Toevaluatean algebraic expression means to find the value of the expression when the variable is replaced by a given number To evaluate an expression, we substitute the given number for the variable in the expression and then simplify the expression using the order of operations

EXAMPLE 2.13 Evaluate x + 7 when

ⓐ x = 3 ⓑ x = 12

Solution

ⓐTo evaluate, substitute 3 for x in the expression, and then simplify

Substitute Add

When x = 3, the expression x + 7 has a value of 10.

ⓑTo evaluate, substitute 12 for x in the expression, and then simplify

Substitute Add

When x = 12, the expression x + 7 has a value of 19.

TRY IT : :2.25 Evaluate:

y + when

ⓐ y = 6 ⓑ y = 15

TRY IT : :2.26 Evaluate:

a − when

ⓐ a = 9 ⓑ a = 17

EXAMPLE 2.14 Evaluate 9x − 2, when

ⓐ x = 5 ⓑ x = 1

Solution

Remember ab means a times b, so 9x means 9 times x.

ⓐTo evaluate the expression when x = 5, we substitute 5 for x, and then simplify

Multiply Subtract

ⓑTo evaluate the expression when x = 1, we substitute 1 for x, and then simplify

Multiply Subtract

Notice that in partⓐthat we wrote 9 ⋅ 5 and in partⓑwe wrote 9(1). Both the dot and the parentheses tell us to multiply

TRY IT : :2.27 Evaluate: 8x − 3, when

TRY IT : :2.28 Evaluate:

4y − 4, when

ⓐ y = 3 ⓑ y = 5

EXAMPLE 2.15

Evaluate x2 when x = 10.

Solution

We substitute 10 for x, and then simplify the expression

Use the definition of exponent Multiply

When x = 10, the expression x2 has a value of 100. TRY IT : :2.29 Evaluate:

x2 when x = 8. TRY IT : :2.30 Evaluate:

x3 when x = 6. EXAMPLE 2.16

Evaluate 2x when x = 5.

Solution

In this expression, the variable is an exponent

Use the definition of exponent Multiply

When x = 5, the expression 2x has a value of 32. TRY IT : :2.31 Evaluate:

2x when x = 6. TRY IT : :2.32 Evaluate:

Evaluate 3x + 4y − when x = 10 and y = 2.

Solution

This expression contains two variables, so we must make two substitutions

Multiply

Add and subtract left to right

When x = 10 and y = 2, the expression 3x + 4y − 6 has a value of 32. TRY IT : :2.33 Evaluate:

2x + 5y − when x = 11 and y = 3 TRY IT : :2.34 Evaluate:

5x − 2y − when x = and y = 8 EXAMPLE 2.18

Evaluate 2x2+ 3x + when x = 4.

Solution

We need to be careful when an expression has a variable with an exponent In this expression, 2x2 means 2 ⋅ x ⋅ x and is different from the expression (2x)2, which means 2x ⋅ 2x.

Simplify 42 Multiply Add

TRY IT : :2.35 Evaluate:

3x2+ 4x + when x = 3. TRY IT : :2.36 Evaluate:

6x2− 4x − when x = 2.

Identify Terms, Coefficients, and Like Terms

The constant that multiplies the variable(s) in a term is called thecoefficient We can think of the coefficient as the numberin front of the variable The coefficient of the term 3x is 3. When we write x, the coefficient is 1, since

x = ⋅ x. Table 2.36gives the coefficients for each of the terms in the left column Term Coefficient

7 7

9a 9

y 1

5x2 5

Table 2.36

An algebraic expression may consist of one or more terms added or subtracted In this chapter, we will only work with terms that are added together.Table 2.37gives some examples of algebraic expressions with various numbers of terms Notice that we include the operation before a term with it

Expression Terms

7 7

y y

x + 7 x, 7

2x + 7y + 4 2x, 7y, 4 3x2+ 4x2+ 5y + 3 3x2, 4x2, 5y, 3

Table 2.37

EXAMPLE 2.19

Identify each term in the expression 9b + 15x2+ a + 6. Then identify the coefficient of each term

Solution

The expression has four terms They are 9b, 15x2, a, and 6. The coefficient of 9b is 9.

The coefficient of 15x2 is 15.

Remember that if no number is written before a variable, the coefficient is 1. So the coefficient of ais 1. The coefficient of a constant is the constant, so the coefficient of 6 is 6.

TRY IT : :2.37 Identify all terms in the given expression, and their coefficients: 4x + 3b + 2

TRY IT : :2.38 Identify all terms in the given expression, and their coefficients:

9a + 13a2+ a3

Which of these terms are like terms?

• The terms 7 and 4 are both constant terms • The terms 5x and 3x are both terms with x.

• The terms n2 and 9n2 both have n2.

Terms are calledlike termsif they have the same variables and exponents All constant terms are also like terms So among the terms 5x, 7, n2, 4, 3x, 9n2,

7 and are like terms. 5x and 3x are like terms.

n2and 9n2are like terms.

Like Terms

Terms that are either constants or have the same variables with the same exponents are like terms

EXAMPLE 2.20 Identify the like terms:

ⓐ y3, 7x2, 14, 23, 4y3, 9x, 5x2 ⓑ 4x2+ 2x + 5x2+ 6x + 40x + 8xy

Solution

ⓐ y3, 7x2, 14, 23, 4y3, 9x, 5x2

Look at the variables and exponents The expression contains y3, x2, x, and constants The terms y3 and 4y3 are like terms because they both have y3.

The terms 7x2 and 5x2 are like terms because they both have x2. The terms 14 and 23 are like terms because they are both constants

The term 9x does not have any like terms in this list since no other terms have the variable xraised to the power of 1.

ⓑ 4x2+ 2x + 5x2+ 6x + 40x + 8xy

Look at the variables and exponents The expression contains the terms 4x2, 2x, 5x2, 6x, 40x, and 8xy The terms 4x2 and 5x2 are like terms because they both have x2.

The terms 2x, 6x, and 40x are like terms because they all have x.

The term 8xy has no like terms in the given expression because no other terms contain the two variables xy. TRY IT : :2.39 Identify the like terms in the list or the expression:

9, 2x3, y2, 8x3, 15, 9y, 11y2

TRY IT : :2.40 Identify the like terms in the list or the expression:

4x3+ 8x2+ 19 + 3x2+ 24 + 6x3

Simplify Expressions by Combining Like Terms

Add the coefficients and keep the same variable It doesn’t matter what x is If you have 3 of something and add 6 more of the same thing, the result is 9 of them For example, 3 oranges plus 6 oranges is 9 oranges We will discuss the mathematical properties behind this later

The expression 3x + 6x has only two terms When an expression contains more terms, it may be helpful to rearrange the terms so that like terms are together The Commutative Property of Addition says that we can change the order of addends without changing the sum So we could rearrange the following expression before combining like terms

Now it is easier to see the like terms to be combined

EXAMPLE 2.21

Simplify the expression: 3x + + 4x + 5.

Solution

Identify the like terms

Rearrange the expression, so the like terms are together Add the coefficients of the like terms

The original expression is simplified to

TRY IT : :2.41 Simplify:

7x + + 9x + 8 TRY IT : :2.42 Simplify:

5y + + 8y + 4y + 5 EXAMPLE 2.22

Simplify the expression: 7x2+ 8x + x2+ 4x.

HOW TO : :COMBINE LIKE TERMS Identify like terms

Rearrange the expression so like terms are together Add the coefficients of the like terms

Identify the like terms

Rearrange the expression so like terms are together Add the coefficients of the like terms

These are not like terms and cannot be combined So 8x2+ 12x is in simplest form TRY IT : :2.43 Simplify:

3x2+ 9x + x2+ 5x TRY IT : :2.44 Simplify:

11y2+ 8y + y2+ 7y

Translate Words to Algebraic Expressions

In the previous section, we listed many operation symbols that are used in algebra, and then we translated expressions and equations into word phrases and sentences Now we’ll reverse the process and translate word phrases into algebraic expressions The symbols and variables we’ve talked about will help us that They are summarized inTable 2.40

Operation Phrase Expression

Addition a plus b

the sum of a and b a increased by b b more than a the total of a and b

b added to a

a + b

Subtraction a minus b

the difference of a and b b subtracted from a a decreased by b b less than a

a − b

Multiplication a times b

the product of a and b a ⋅ b, ab, a(b), (a)(b)

Division a divided by b

the quotient of a and b the ratio of a and b

b divided into a

a ÷ b, a / b, ab, b a

Table 2.40

Look closely at these phrases using the four operations: • the sumof a and b

• the quotientof a and b

Each phrase tells you to operate on two numbers Look for the wordsofandandto find the numbers EXAMPLE 2.23

Translate each word phrase into an algebraic expression:

ⓐthe difference of 20 and 4 ⓑthe quotient of 10x and 3

Solution

ⓐThe key word isdifference, which tells us the operation is subtraction Look for the wordsofandandto find the numbers to subtract

the diffe ence of 20 and 4 20 minus 4

20 − 4

ⓑThe key word isquotient, which tells us the operation is division the quotient of 10x and 3

divide 10x by 3 10x ÷ 3

This can also be written as 10x / or 10x3

TRY IT : :2.45 Translate the given word phrase into an algebraic expression:

ⓐthe difference of 47 and 41 ⓑthe quotient of5x and 2 TRY IT : :2.46 Translate the given word phrase into an algebraic expression:

ⓐthe sum of 17 and 19 ⓑthe product of 7 and x

How old will you be in eight years? What age is eight more years than your age now? Did you add 8 to your present age? Eightmore thanmeans eight added to your present age

How old were you seven years ago? This is seven years less than your age now You subtract 7 from your present age Sevenless thanmeans seven subtracted from your present age

EXAMPLE 2.24

Translate each word phrase into an algebraic expression:

ⓐEight more than y ⓑSeven less than9z

Solution

ⓐThe key words aremore than They tell us the operation is addition.More thanmeans “added to” Eight more than y

Eight added to y

y + 8

ⓑThe key words areless than They tell us the operation is subtraction.Less thanmeans “subtracted from” Seven less than 9z

ⓐEleven more than x ⓑFourteen less than 11a TRY IT : :2.48 Translate each word phrase into an algebraic expression:

ⓐ19 more than j ⓑ 21 less than 2x EXAMPLE 2.25

Translate each word phrase into an algebraic expression:

ⓐfive times the sum of m and n ⓑthe sum of five times m and n

Solution

ⓐThere are two operation words:timestells us to multiply andsumtells us to add Because we are multiplying 5 times the sum, we need parentheses around the sum of m and n.

five times the sum of m and n

5(m + n)

ⓑTo take a sum, we look for the wordsofandandto see what is being added Here we are taking the sumoffive times

m and n.

the sum of five times m and n

5m + n

Notice how the use of parentheses changes the result In partⓐ, we add first and in partⓑ, we multiply first TRY IT : :2.49 Translate the word phrase into an algebraic expression:

ⓐfour times the sum of p and q ⓑthe sum of four times p and q TRY IT : :2.50 Translate the word phrase into an algebraic expression:

ⓐthe difference of two times x and 8 ⓑtwo times the difference of x and 8 Later in this course, we’ll apply our skills in algebra to solving equations We’ll usually start by translating a word phrase to an algebraic expression We’ll need to be clear about what the expression will represent We’ll see how to this in the next two examples

EXAMPLE 2.26

Solution

Write a phrase about the height 6 less than the width Substitute w for the width 6 less than w Rewrite 'less than' as 'subtracted from' 6 subtracted from w Translate the phrase into algebra w − 6

TRY IT : :2.51

The length of a rectangle is 5 inches less than the width Let w represent the width of the rectangle Write an expression for the length of the rectangle

TRY IT : :2.52

The width of a rectangle is 2 meters greater than the length Let l represent the length of the rectangle Write an expression for the width of the rectangle

EXAMPLE 2.27

Blanca has dimes and quarters in her purse The number of dimes is 2 less than 5 times the number of quarters Let q represent the number of quarters Write an expression for the number of dimes

Solution

Write a phrase about the number of dimes two less than five times the number of quarters Substitute q for the number of quarters 2 less than five times q

Translate 5 times q 2 less than 5q

Translate the phrase into algebra 5q − 2

TRY IT : :2.53

Geoffrey has dimes and quarters in his pocket The number of dimes is seven less than six times the number of quarters Let q represent the number of quarters Write an expression for the number of dimes

TRY IT : :2.54

Lauren has dimes and nickels in her purse The number of dimes is eight more than four times the number of nickels Let n represent the number of nickels Write an expression for the number of dimes

MEDIA : :ACCESS ADDITIONAL ONLINE RESOURCES

Practice Makes Perfect

Evaluate Algebraic Expressions

In the following exercises, evaluate the expression for the given value.

69. 7x + when x = 2 70. 9x + when x = 3 71. 5x − when x = 6 72. 8x − when x = 7 73. x2 when x = 12 74. x3when x = 5

75. x5when x = 2 76. x4 when x = 3 77. 3xwhen x = 3

78. 4x when x = 2 79. x2+ 3x − when x = 4 80. x2+ 5x − when x = 6 81.

2x + 4y − when x = 7, y = 8 82.6x + 3y − when x = 6, y = 9 83. (x − y)2when x = 10, y = 7 84. (x + y)2when x = 6, y = 9 85. a2+ b2when a = 3, b = 8 86. r2− s2when r = 12, s = 5 87.

2l + 2w when l = 15, w = 12 88.2l + 2w when l = 18, w = 14

Identify Terms, Coefficients, and Like Terms

In the following exercises, list the terms in the given expression.

89. 15x2+ 6x + 2 90. 11x2+ 8x + 5 91. 10y3+ y + 2

92. 9y3+ y + 5

In the following exercises, identify the coefficient of the given term.

93. 8a 94. 13m 95. 5r2

96. 6x3

In the following exercises, identify all sets of like terms.

97. x3, 8x, 14, 8y, 5, 8x3 98.6z, 3w2, 1, 6z2, 4z, w2 99.

9a, a2, 16ab, 16b2, 4ab, 9b2 100. 3, 25r2, 10s, 10r, 4r2, 3s

Simplify Expressions by Combining Like Terms

In the following exercises, simplify the given expression by combining like terms.

101. 10x + 3x 102. 15x + 4x 103. 17a + 9a

104. 18z + 9z 105. 4c + 2c + c 106. 6y + 4y + y

110. 8d + + 2d + 5 111.7p + + 5p + 4 112. 8x + + 4x − 5 113. 10a + + 5a − + 7a − 4 114. 7c + + 6c − + 9c − 1 115.

3x2+ 12x + 11 + 14x2+ 8x + 5 116.

5b2+ 9b + 10 + 2b2+ 3b − 4

Translate English Phrases into Algebraic Expressions

In the following exercises, translate the given word phrase into an algebraic expression.

117.The sum of and 12 118.The sum of and 119.The difference of 14 and 120.8 less than 19 121.The product of and 122.The product of and 123.The quotient of 36 and 124.The quotient of 42 and 125.The difference of x and 4 126. 3 less than x 127.The product of 6 and y 128.The product of 9 and y

129.The sum of 8x and 3x 130.The sum of 13x and 3x 131.The quotient of y and 3 132.The quotient of y and 8 133.Eight times the difference of

y and nine 134.yand oneSeven times the difference of 135.Five times the sum of x and

y 136.x Nine times five less than twice

In the following exercises, write an algebraic expression.

137. Adele bought a skirt and a blouse The skirt cost $15 more than the blouse Let b represent the cost of the blouse Write an expression for the cost of the skirt

138.Eric has rock and classical CDs in his car The number of rock CDs is 3 more than the number of classical CDs Let c represent the number of classical CDs Write an expression for the number of rock CDs

139. The number of girls in a second-grade class is 4 less than the number of boys Let b

represent the number of boys Write an expression for the number of girls

140. Marcella has 6 fewer male cousins than female cousins Let

f represent the number of female cousins Write an expression for the number of boy cousins

141.Greg has nickels and pennies in his pocket The number of pennies is seven less than twice the number of nickels Let n

represent the number of nickels Write an expression for the number of pennies

142. Jeannette has $5 and $10 bills in her wallet The number of fives is three more than six times the number of tens Let t represent the number of tens Write an expression for the number of fives

Everyday Math

In the following exercises, use algebraic expressions to solve the problem.

143.Car insuranceJustin’s car insurance has a $750 deductible per incident This means that he pays $750 and his insurance company will pay all costs beyond

$750. If Justin files a claim for $2,100, how much will he pay, and how much will his insurance company pay?

145. Explain why “the sum of x and y” is the same as “the sum ofyandx,” but “the difference ofxand y” is not the same as “the difference ofyandx.” Try substituting two random numbers for xand yto help you explain

146.Explain the difference between “4 times the sum of x and y” and “the sum of 4times x and y.”

Self Check

ⓐAfter completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

2.3 Solving Equations Using the Subtraction and Addition Properties of Equality

Learning Objectives

By the end of this section, you will be able to:

Determine whether a number is a solution of an equation Model the Subtraction Property of Equality

Solve equations using the Subtraction Property of Equality Solve equations using the Addition Property of Equality Translate word phrases to algebraic equations

Translate to an equation and solve

Be Prepared!

Before you get started, take this readiness quiz Evaluate x + when x = 11.

If you missed this problem, reviewExample 2.13 Evaluate 5x − when x = 9.

If you missed this problem, reviewExample 2.14 Translate into algebra: the difference of x and 8.

If you missed this problem, reviewExample 2.24

When some people hear the wordalgebra, they think of solving equations The applications of solving equations are limitless and extend to all careers and fields In this section, we will begin solving equations We will start by solving basic equations, and then as we proceed through the course we will build up our skills to cover many different forms of equations

Determine Whether a Number is a Solution of an Equation

Solving an equation is like discovering the answer to a puzzle An algebraic equation states that two algebraic expressions are equal To solve an equation is to determine the values of the variable that make the equation a true statement Any number that makes the equation true is called asolutionof the equation It is the answer to the puzzle!

Solution of an Equation

Asolution to an equationis a value of a variable that makes a true statement when substituted into the equation The process of finding the solution to an equation is called solving the equation

To find the solution to an equation means to find the value of the variable that makes the equation true Can you recognize the solution of x + = 7? If you said5, you’re right! We say5 is a solution to the equation x + = 7 because when we substitute 5 for x the resulting statement is true

x + = 7

5 + =? 7 7 = ✓

Since 5 + = 7 is a true statement, we know that 5 is indeed a solution to the equation

EXAMPLE 2.28

Determine whether x = is a solution of 6x − 17 = 16.

Solution

Multiply Subtract

So x = 5 is not a solution to the equation 6x − 17 = 16. TRY IT : :2.55 Is x = a solution of 4x − = 16?

TRY IT : :2.56 Is x = a solution of 6x − = 10?

EXAMPLE 2.29

Determine whether y = is a solution of 6y − = 5y − 2.

Solution

Here, the variable appears on both sides of the equation We must substitute 2 for each y.

Multiply Subtract

Since y = 2 results in a true equation, we know that 2is a solution to the equation 6y − = 5y − 2. TRY IT : :2.57 Is y = a solution of 9y − = 8y + 1?

TRY IT : :2.58 Is y = a solution of 5y − = 3y + 5?

HOW TO : :DETERMINE WHETHER A NUMBER IS A SOLUTION TO AN EQUATION Substitute the number for the variable in the equation

Simplify the expressions on both sides of the equation Determine whether the resulting equation is true

◦ If it is true, the number is a solution ◦ If it is not true, the number is not a solution Step

Model the Subtraction Property of Equality

We will use a model to help you understand how the process of solving an equation is like solving a puzzle An envelope represents the variable – since its contents are unknown – and each counter represents one

Suppose a desk has an imaginary line dividing it in half We place three counters and an envelope on the left side of desk, and eight counters on the right side of the desk as inFigure 2.3 Both sides of the desk have the same number of counters, but some counters are hidden in the envelope Can you tell how many counters are in the envelope?

Figure 2.3

What steps are you taking in your mind to figure out how many counters are in the envelope? Perhaps you are thinking “I need to remove the 3 counters from the left side to get the envelope by itself Those 3 counters on the left match with 3 on the right, so I can take them away from both sides That leaves five counters on the right, so there must be 5 counters in the envelope.”Figure 2.4shows this process

Figure 2.4

What algebraic equation is modeled by this situation? Each side of the desk represents an expression and the center line takes the place of the equal sign We will call the contents of the envelope x, so the number of counters on the left side of the desk is x + 3. On the right side of the desk are 8 counters We are told that x + 3 is equal to 8 so our equation isx + = 8.

Figure 2.5

x + = 8

Let’s write algebraically the steps we took to discover how many counters were in the envelope

First, we took away three from each side Then we were left with five

MANIPULATIVE MATHEMATICS

Doing the Manipulative Mathematics activity, “Subtraction Property of Equality” will help you develop a better understanding of how to solve equations by using the Subtraction Property of Equality

EXAMPLE 2.30

Write an equation modeled by the envelopes and counters, and then solve the equation:

Solution

On the left, write x for the contents of the envelope, add the 4 counters, so we have x + 4 x + 4

On the right, there are 5 counters 5

The two sides are equal x + = 5

Solve the equation by subtracting 4 counters from each side

We can see that there is one counter in the envelope This can be shown algebraically as:

Substitute 1 for xin the equation to check

Since x = 1 makes the statement true, we know that 1 is indeed a solution

TRY IT : :2.60 Write the equation modeled by the envelopes and counters, and then solve the equation:

Solve Equations Using the Subtraction Property of Equality

Our puzzle has given us an idea of what we need to to solve an equation The goal is to isolate the variable by itself on one side of the equations In the previous examples, we used the Subtraction Property of Equality, which states that when we subtract the same quantity from both sides of an equation, we still have equality

Subtraction Property of Equality For any numbers a, b, and c, if

a = b

then

a − c = b − c

Think about twin brothers Andy and Bobby They are 17 years old How old was Andy 3 years ago? He was 3 years less than 17, so his age was 17 − 3, or 14. What about Bobby’s age 3years ago? Of course, he was 14 also Their ages are equal now, and subtracting the same quantity from both of them resulted in equal ages 3 years ago

a = b a − = b − 3

EXAMPLE 2.31 Solve: x + = 17.

Solution

We will use the Subtraction Property of Equality to isolate x.

Subtract from both sides Simplify

Since x = 9 makes x + = 17 a true statement, we know9 is the solution to the equation

HOW TO : :SOLVE AN EQUATION USING THE SUBTRACTION PROPERTY OF EQUALITY Use the Subtraction Property of Equality to isolate the variable

Simplify the expressions on both sides of the equation Check the solution

x + = 19

TRY IT : :2.62 Solve:

x + = 14

EXAMPLE 2.32 Solve: 100 = y + 74.

Solution

To solve an equation, we must always isolate the variable—it doesn’t matter which side it is on To isolate y, we will subtract 74 from both sides

Subtract 74 from both sides Simplify

Substitute 26 for y to check

Since y = 26 makes 100 = y + 74 a true statement, we have found the solution to this equation TRY IT : :2.63 Solve:

95 = y + 67 TRY IT : :2.64 Solve:

91 = y + 45

Solve Equations Using the Addition Property of Equality

In all the equations we have solved so far, a number was added to the variable on one side of the equation We used subtraction to “undo” the addition in order to isolate the variable

But suppose we have an equation with a number subtracted from the variable, such as x − = 8. We want to isolate the variable, so to “undo” the subtraction we will add the number to both sides

We use the Addition Property of Equality, which says we can add the same number to both sides of the equation without changing the equality Notice how it mirrors the Subtraction Property of Equality

Addition Property of Equality For any numbers a, b, and c, if

a = b

then

a = b a + 10 = b + 10

We can add the same number to both sides and still keep the equality

EXAMPLE 2.33 Solve: x − = 8.

Solution

We will use the Addition Property of Equality to isolate the variable

Add to both sides Simplify

TRY IT : :2.65 Solve:

x − = 13

TRY IT : :2.66 Solve:

y − = 3

EXAMPLE 2.34 Solve: 27 = a − 16.

Solution

We will add 16 to each side to isolate the variable

HOW TO : :SOLVE AN EQUATION USING THE ADDITION PROPERTY OF EQUALITY Use the Addition Property of Equality to isolate the variable

Simplify the expressions on both sides of the equation Check the solution

Add 16 to each side Simplify

The solution to 27 = a − 16 is a = 43.

TRY IT : :2.67 Solve:

19 = a − 18 TRY IT : :2.68 Solve:

27 = n − 14

Translate Word Phrases to Algebraic Equations

Remember, an equation has an equal sign between two algebraic expressions So if we have a sentence that tells us that two phrases are equal, we can translate it into an equation We look for clue words that meanequals Some words that translate to the equal sign are:

• is equal to • is the same as • is

• gives • was • will be

It may be helpful to put a box around theequalsword(s) in the sentence to help you focus separately on each phrase Then translate each phrase into an expression, and write them on each side of the equal sign

We will practice translating word sentences into algebraic equations Some of the sentences will be basic number facts with no variables to solve for Some sentences will translate into equations with variables The focus right now is just to translate the words into algebra

EXAMPLE 2.35

Translate the sentence into an algebraic equation: The sum of 6 and 9 is 15.

Solution

The wordistells us the equal sign goes between and 15

Locate the “equals” word(s) Write the = sign

Translate the words to the left of theequalsword into an algebraic expression

TRY IT : :2.69 Translate the sentence into an algebraic equation:

The sum of 7 and 6gives 13.

TRY IT : :2.70 Translate the sentence into an algebraic equation: The sum of 8 and 6is 14.

EXAMPLE 2.36

Translate the sentence into an algebraic equation: The product of 8 and 7 is 56.

Solution

The location of the wordistells us that the equal sign goes between and 56

Locate the “equals” word(s) Write the = sign

Translate the words to the left of theequalsword into an algebraic expression

Translate the words to the right of theequalsword into an algebraic expression

TRY IT : :2.71 Translate the sentence into an algebraic equation: The product of 6 and 9 is 54.

TRY IT : :2.72 Translate the sentence into an algebraic equation:

The product of 21 and 3 gives 63. EXAMPLE 2.37

Translate the sentence into an algebraic equation: Twice the difference of x and 3 gives 18.

Solution

Locate the “equals” word(s)

Recognize the key words:twice; difference of … and … Twicemeans two times Translate

TRY IT : :2.73 Translate the given sentence into an algebraic equation:

Twice the difference of x and 5 gives 30.

Now let’s practice translating sentences into algebraic equations and then solving them We will solve the equations by using the Subtraction and Addition Properties of Equality

EXAMPLE 2.38

Translate and solve: Three more than x is equal to 47.

Solution

Three more thanxis equal to 47 Translate

Subtract from both sides of the equation Simplify

We can check Let x = 44

So x = 44 is the solution

TRY IT : :2.75 Translate and solve:

Seven more than x is equal to 37. TRY IT : :2.76 Translate and solve:

Eleven more than y is equal to 28. EXAMPLE 2.39

Translate and solve: The difference of y and 14 is 18.

Solution

The difference ofyand 14 is 18 Translate

Add 14 to both sides Simplify

We can check Let y = 32

So y = 32 is the solution

TRY IT : :2.78 Translate and solve:

The difference of x and 19 is equal to 45. MEDIA : :ACCESS ADDITIONAL ONLINE RESOURCES

Practice Makes Perfect

Determine Whether a Number is a Solution of an Equation

In the following exercises, determine whether each given value is a solution to the equation.

147. x + 13 = 21

ⓐ x = 8 ⓑ x = 34

148. y + 18 = 25

ⓐ y = 7 ⓑy = 43

149. m − = 13

ⓐ m = 9 ⓑm = 17

150. n − = 6

ⓐ n = 3 ⓑn = 15

151.3p + = 15

ⓐ p = 3 ⓑ p = 7

152. 8q + = 20

ⓐ q = 2 ⓑ q = 3

153. 18d − = 27

ⓐ d = 1 ⓑ d = 2

154. 24 f − 12 = 60

ⓐ f = 2 ⓑ f = 3

155. 8u − = 4u + 40

ⓐ u = 3 ⓑu = 11

156. 7v − = 4v + 36

ⓐ v = 3 ⓑv = 11

157. 20h − = 15h + 35

ⓐ h = 6 ⓑh = 8

158. 18k − = 12k + 33

ⓐ k = 1 ⓑ k = 6

Model the Subtraction Property of Equality

In the following exercises, write the equation modeled by the envelopes and counters and then solve using the subtraction property of equality.

159. 160. 161.

162.

Solve Equations using the Subtraction Property of Equality

In the following exercises, solve each equation using the subtraction property of equality.

163. a + = 18 164. b + = 13 165. p + 18 = 23 166. q + 14 = 31 167. r + 76 = 100 168. s + 62 = 95

169. 16 = x + 9 170. 17 = y + 6 171. 93 = p + 24

172. 116 = q + 79 173. 465 = d + 398 174. 932 = c + 641

Solve Equations using the Addition Property of Equality

In the following exercises, solve each equation using the addition property of equality.

175. y − = 19 176. x − = 12 177. u − = 24 178. v − = 35 179. f − 55 = 123 180. g − 39 = 117

181. 19 = n − 13 182. 18 = m − 15 183. 10 = p − 38

184. 18 = q − 72 185. 268 = y − 199 186. 204 = z − 149

Translate Word Phrase to Algebraic Equations

In the following exercises, translate the given sentence into an algebraic equation.

187.The sum of 8 and 9 is equal to 17.

188.The sum of 7 and 9 is equal to 16.

189.The difference of 23 and 19 is equal to 4.

190.The difference of 29 and 12 is equal to 17.

191.The product of 3 and 9 is equal to 27.

192. The product of 6 and 8 is equal to 48.

193.The quotient of 54 and 6 is equal to 9.

194.The quotient of 42 and 7 is equal to 6.

195.Twice the difference of n and 10 gives 52.

196. Twice the difference of m

and 14 gives 64.

197.The sum of three times yand 10 is 100.

198.The sum of eight times x and 4 is 68.

Translate to an Equation and Solve

In the following exercises, translate the given sentence into an algebraic equation and then solve it.

199.Five more than p is equal to 21.

200.Nine more than q is equal to 40.

201.The sum of r and 18 is 73. 202.The sum of s and 13 is 68. 203.The difference of d and 30

is equal to 52.

204.The difference of c and 25 is equal to 75.

205. 12 less than u is 89. 206. 19 less than w is 56. 207. 325 less than c gives 799. 208. 299 less than d gives 850.

Everyday Math

209. Insurance Vince’s car insurance has a $500 deductible Find the amount the insurance company will pay, p, for an $1800 claim by solving the equation 500 + p = 1800.

210.InsuranceMarta’s homeowner’s insurance policy has a $750 deductible The insurance company paid $5800 to repair damages caused by a storm Find the total cost of the storm damage, d, by solving the equation d − 750 = 5800.

211.Sale purchaseArthur bought a suit that was on sale for $120 off He paid $340 for the suit Find the original price, p, of the suit by solving the equation

p − 120 = 340.

213. Is x = 1 a solution to the equation 8x − = 16 − 6x? How you know?

214. Write the equation y − = 21 in words Then make up a word problem for this equation

Self Check

ⓐAfter completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

2.4 Find Multiples and Factors Learning Objectives

By the end of this section, you will be able to:

Identify multiples of numbers Use common divisibility tests Find all the factors of a number Identify prime and composite numbers

Be Prepared!

Before you get started, take this readiness quiz

1 Which of the following numbers are counting numbers (natural numbers)?

0, 4, 215

If you missed this problem, reviewExample 1.1 Find the sum of 3, 5, and 7.

If you missed the problem, reviewExample 2.1

Identify Multiples of Numbers

Annie is counting the shoes in her closet The shoes are matched in pairs, so she doesn’t have to count each one She counts by twos: 2, 4, 6, 8, 10, 12. She has 12 shoes in her closet

The numbers 2, 4, 6, 8, 10, 12 are called multiples of 2. Multiples of 2 can be written as the product of a counting number and 2. The first six multiples of 2 are given below

1 ⋅ = 2 2 ⋅ = 4 3 ⋅ = 6 4 ⋅ = 8 5 ⋅ = 10 6 ⋅ = 12

Amultiple of a numberis the product of the number and a counting number So a multiple of 3 would be the product of a counting number and 3. Below are the first six multiples of 3.

1 ⋅ = 3 2 ⋅ = 6 3 ⋅ = 9 4 ⋅ = 12 5 ⋅ = 15 6 ⋅ = 18

Counting Number 1 2 3 4 5 6 7 8 9 10 11 12

Multiples of 2 2 4 6 8 10 12 14 16 18 20 22 24

Multiples of 3 3 6 9 12 15 18 21 24 27 30 33 36

Multiples of 4 4 8 12 16 20 24 28 32 36 40 44 48

Multiples of 5 5 10 15 20 25 30 35 40 45 50 55 60

Multiples of 6 6 12 18 24 30 36 42 48 54 60 66 72

Multiples of 7 7 14 21 28 35 42 49 56 63 70 77 84

Multiples of 8 8 16 24 32 40 48 56 64 72 80 88 96

Multiples of 9 9 18 27 36 45 54 63 72 81 90 99 108

Table 2.55

Multiple of a Number

A number is a multiple of nif it is the product of a counting number and n.

Recognizing the patterns for multiples of 2, 5, 10, and 3 will be helpful to you as you continue in this course MANIPULATIVE MATHEMATICS

Doing the Manipulative Mathematics activity “Multiples” will help you develop a better understanding of multiples

Figure 2.6shows the counting numbers from 1 to 50. Multiples of 2are highlighted Do you notice a pattern?

Figure 2.6 Multiples of 2between 1and 50

The last digit of each highlighted number inFigure 2.6is either 0, 2, 4, 6, or 8. This is true for the product of 2 and any counting number So, to tell if any number is a multiple of 2 look at the last digit If it is 0, 2, 4, 6, or 8, then the number is a multiple of 2.

EXAMPLE 2.40

Determine whether each of the following is a multiple of 2:

Solution

ⓐ

Is 489 a multiple of 2?

Is the last digit 0, 2, 4, 6, or 8? No

489 is not a multiple of

ⓑ

Is 3,714 a multiple of 2?

Is the last digit 0, 2, 4, 6, or 8? Yes

3,714 is a multiple of

TRY IT : :2.79 Determine whether each number is a multiple of 2:

ⓐ 678 ⓑ 21,493

TRY IT : :2.80 Determine whether each number is a multiple of 2:

ⓐ 979 ⓑ 17,780

Now let’s look at multiples of 5. Figure 2.7highlights all of the multiples of 5 between 1 and 50. What you notice about the multiples of 5?

Figure 2.7 Multiples of 5between 1and 50

All multiples of 5 end with either 5 or 0. Just like we identify multiples of 2 by looking at the last digit, we can identify multiples of 5 by looking at the last digit

EXAMPLE 2.41

Determine whether each of the following is a multiple of 5:

ⓐ

Is 579 a multiple of 5? Is the last digit or 0? No

579 is not a multiple of

ⓑ

Is 880 a multiple of 5? Is the last digit or 0? Yes

880 is a multiple of

TRY IT : :2.81 Determine whether each number is a multiple of 5.

ⓐ 675 ⓑ 1,578

TRY IT : :2.82 Determine whether each number is a multiple of 5.

ⓐ 421 ⓑ 2,690

Figure 2.8highlights the multiples of 10 between 1 and 50.All multiples of 10 all end with a zero

Figure 2.8 Multiples of 10between 1 and 50

EXAMPLE 2.42

Determine whether each of the following is a multiple of 10:

ⓐ 425 ⓑ 350

Solution

ⓐ

Is 425 a multiple of 10? Is the last digit zero? No

ⓑ

Is 350 a multiple of 10? Is the last digit zero? Yes

350 is a multiple of 10

TRY IT : :2.83 Determine whether each number is a multiple of 10:

ⓐ 179 ⓑ 3,540

TRY IT : :2.84 Determine whether each number is a multiple of 10:

ⓐ 110 ⓑ 7,595

Figure 2.9highlights multiples of 3. The pattern for multiples of 3 is not as obvious as the patterns for multiples of 2, 5, and 10.

Figure 2.9 Multiples of 3between 1and 50

Unlike the other patterns we’ve examined so far, this pattern does not involve the last digit The pattern for multiples of 3 is based on the sum of the digits If the sum of the digits of a number is a multiple of 3, then the number itself is a multiple of 3. SeeTable 2.62

Multiple of 3 3 6 9 12 15 18 21 24

Sum of digits 3 6 9 1 + 23 1 + 56 1 + 89 2 + 13 2 + 46

Table 2.62

Consider the number 42. The digits are 4 and 2, and their sum is 4 + = 6. Since 6 is a multiple of 3, we know that 42 is also a multiple of 3.

EXAMPLE 2.43

Determine whether each of the given numbers is a multiple of 3:

ⓐ 645 ⓑ 10,519

Solution

Find the sum of the digits 6 + + = 15

Is 15 a multiple of 3? Yes

If we're not sure, we could add its digits to find out We can check it by dividing 645 by 645 ÷ 3

The quotient is 215 3 ⋅ 215 = 645

ⓑIs 10,519 a multiple of 3?

Find the sum of the digits 1 + + + + = 16 Is 16 a multiple of 3? No

So 10,519 is not a multiple of either 645 ÷ 3

We can check this by dividing by 10,519 by 3 10,5193,506R1

When we divide 10,519 by 3, we not get a counting number, so 10,519 is not the product of a counting number and 3. It is not a multiple of 3.

TRY IT : :2.85 Determine whether each number is a multiple of 3:

ⓐ 954 ⓑ 3,742

TRY IT : :2.86 Determine whether each number is a multiple of 3:

ⓐ 643 ⓑ 8,379

Look back at the charts where you highlighted the multiples of 2, of 5, and of 10. Notice that the multiples of 10 are the numbers that are multiples of both 2 and 5. That is because 10 = ⋅ 5. Likewise, since 6 = ⋅ 3, the multiples of 6 are the numbers that are multiples of both 2 and 3.

Use Common Divisibility Tests

Another way to say that 375 is a multiple of 5 is to say that 375 is divisible by 5. In fact, 375 ÷ 5 is 75, so 375 is 5 ⋅ 75. Notice inExample 2.43that 10,519 is not a multiple 3. When we divided 10,519 by 3 we did not get a counting number, so 10,519 is not divisible by 3.

Divisibility

If a number m is a multiple of n, then we say that m is divisible by n.

Divisibility Tests A number is divisible by

2 if the last digit is 0, 2, 4, 6, or 8 3 if the sum of the digits is divisible by 3 5 if the last digit is 5 or 0 6 if divisible by both 2 and 3 10 if the last digit is 0 Table 2.65

EXAMPLE 2.44

Determine whether 1,290 is divisible by 2, 3, 5, and 10.

Solution

Table 2.66applies the divisibility tests to 1,290. In the far right column, we check the results of the divisibility tests by seeing if the quotient is a whole number

Divisible by…? Test Divisible? Check

2 Is last digit 0, 2, 4, 6, or 8? Yes. yes 1290 ÷ = 645 3 Is sum of digits divisible by 3?1 + + + = 12 Yes. yes 1290 ÷ = 430 5 Is last digit 5 or 0? Yes. yes 1290 ÷ = 258 10 Is last digit 0? Yes. yes 1290 ÷ 10 = 129

Table 2.66

Thus, 1,290 is divisible by 2, 3, 5, and 10.

TRY IT : :2.87 Determine whether the given number is divisible by 2, 3, 5, and 10.

6240

TRY IT : :2.88 Determine whether the given number is divisible by 2, 3, 5, and 10.

7248 EXAMPLE 2.45

Determine whether 5,625 is divisible by 2, 3, 5, and 10.

Solution

Divisible by…? Test Divisible? Check

2 Is last digit 0, 2, 4, 6, or 8? No. no 5625 ÷ = 2812.5

3 Is sum of digits divisible by 3?5 + + + = 18 Yes. yes 5625 ÷ = 1875 5 Is last digit is 5 or 0? Yes. yes 5625 ÷ = 1125 10 Is last digit 0? No. no 5625 ÷ 10 = 562.5 Table 2.67

Thus, 5,625 is divisible by 3 and 5, but not 2, or 10.

TRY IT : :2.89 Determine whether the given number is divisible by 2, 3, 5, and 10. 4962

TRY IT : :2.90 Determine whether the given number is divisible by 2, 3, 5, and 10. 3765

Find All the Factors of a Number

There are often several ways to talk about the same idea So far, we’ve seen that if m is a multiple of n, we can say that

m is divisible by n. We know that 72 is the product of 8 and 9, so we can say 72 is a multiple of 8 and 72 is a multiple of 9. We can also say 72 is divisible by 8 and by 9. Another way to talk about this is to say that 8 and 9 are factors of 72. When we write 72 = ⋅ 9 we can say that we have factored 72.

Factors

If a ⋅ b = m, then a and b are factors of m, and m is the product of a and b.

In algebra, it can be useful to determine all of the factors of a number This is called factoring a number, and it can help us solve many kinds of problems

MANIPULATIVE MATHEMATICS

Doing the Manipulative Mathematics activity “Model Multiplication and Factoring” will help you develop a better understanding of multiplication and factoring

For example, suppose a choreographer is planning a dance for a ballet recital There are 24 dancers, and for a certain scene, the choreographer wants to arrange the dancers in groups of equal sizes on stage

Number of Groups Dancers per Group Total Dancers

1 24 1 ⋅ 24 = 24

2 12 2 ⋅ 12 = 24

3 8 3 ⋅ = 24

4 6 4 ⋅ = 24

6 4 6 ⋅ = 24

8 3 8 ⋅ = 24

12 2 12 ⋅ = 24

24 1 24 ⋅ = 24

Table 2.68

What patterns you see inTable 2.68? Did you notice that the number of groups times the number of dancers per group is always 24? This makes sense, since there are always 24 dancers

You may notice another pattern if you look carefully at the first two columns These two columns contain the exact same set of numbers—but in reverse order They are mirrors of one another, and in fact, both columns list all of the factors of

24, which are:

1, 2, 3, 4, 6, 8, 12, 24

We can find all the factors of any counting number by systematically dividing the number by each counting number, starting with 1. If the quotient is also a counting number, then the divisor and the quotient are factors of the number We can stop when the quotient becomes smaller than the divisor

EXAMPLE 2.46

Find all the factors of 72.

Solution

Divide 72 by each of the counting numbers starting with 1. If the quotient is a whole number, the divisor and quotient are a pair of factors

HOW TO : :FIND ALL THE FACTORS OF A COUNTING NUMBER

Divide the number by each of the counting numbers, in order, until the quotient is smaller than the divisor

◦ If the quotient is a counting number, the divisor and quotient are a pair of factors ◦ If the quotient is not a counting number, the divisor is not a factor

List all the factor pairs

Write all the factors in order from smallest to largest Step

The next line would have a divisor of 9 and a quotient of 8. The quotient would be smaller than the divisor, so we stop If we continued, we would end up only listing the same factors again in reverse order Listing all the factors from smallest to greatest, we have

1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, and 72

TRY IT : :2.91 Find all the factors of the given number:

96

TRY IT : :2.92 Find all the factors of the given number: 80

Identify Prime and Composite Numbers

Some numbers, like 72, have many factors Other numbers, such as 7, have only two factors: 1 and the number A number with only two factors is called aprime number A number with more than two factors is called acomposite number The number 1 is neither prime nor composite It has only one factor, itself

Prime Numbers and Composite Numbers

A prime number is a counting number greater than 1 whose only factors are 1 and itself A composite number is a counting number that is not prime

Figure 2.10lists the counting numbers from2 through 20 along with their factors The highlighted numbers are prime, since each has only two factors

Figure 2.10 Factors of the counting numbers from 2through 20, with prime numbers highlighted

EXAMPLE 2.47

Identify each number as prime or composite:

ⓐ 83 ⓑ 77

Solution

ⓐTest each prime, in order, to see if it is a factor of 83, starting with 2, as shown We will stop when the quotient is smaller than the divisor

Prime Test Factor of 83?

2 Last digit of 83 is not 0, 2, 4, 6, or 8. No

3 8 + = 11, and 11 is not divisible by 3. No

5 The last digit of 83 is not 5 or 0. No

7 83 ÷ = 11.857…. No

11 83 ÷ 11 = 7.545… No

We can stop when we get to 11 because the quotient (7.545…) is less than the divisor We did not find any prime numbers that are factors of 83, so we know 83 is prime

ⓑTest each prime, in order, to see if it is a factor of 77.

Prime Test Factor of 77?

2 Last digit is not 0, 2, 4, 6, or 8. No

3 7 + = 14, and 14 is not divisible by 3. No

5 the last digit is not 5 or 0. No

7 77 ÷ 11 = 7 Yes

Since 77 is divisible by 7, we know it is not a prime number It is composite

TRY IT : :2.93 Identify the number as prime or composite:

91

HOW TO : :DETERMINE IF A NUMBER IS PRIME

Test each of the primes, in order, to see if it is a factor of the number

Start with 2 and stop when the quotient is smaller than the divisor or when a prime factor is found

If the number has a prime factor, then it is a composite number If it has no prime factors, then the number is prime

137 LINKS TO LITERACY

The Links to Literacy activitiesOne Hundred Hungry Ants,Spunky Monkeys on ParadeandA Remainder of Onewill provide you with another view of the topics covered in this section

MEDIA : :ACCESS ADDITIONAL ONLINE RESOURCES

• Divisibility Rules (http://openstaxcollege.org/l/24Divisrules)

• Factors (http://openstaxcollege.org/l/24Factors)

• Ex 1: Determine Factors of a Number (http://openstaxcollege.org/l/24Factors1)

• Ex 2: Determine Factors of a Number (http://openstaxcollege.org/l/24Factors2)

Practice Makes Perfect

Identify Multiples of Numbers

In the following exercises, list all the multiples less than 50 for the given number.

215. 2 216. 3 217. 4

218. 5 219. 6 220. 7

221. 8 222.9 223. 10

224. 12

Use Common Divisibility Tests

In the following exercises, use the divisibility tests to determine whether each number is divisible by 2, 3, 4, 5, 6, and 10.

225. 84 226.96 227. 75

228. 78 229.168 230. 264

231. 900 232.800 233. 896

234. 942 235. 375 236. 750

237. 350 238. 550 239. 1430

240. 1080 241. 22,335 242. 39,075

Find All the Factors of a Number

In the following exercises, find all the factors of the given number.

243. 36 244. 42 245. 60

246. 48 247.144 248. 200

249. 588 250.576

Identify Prime and Composite Numbers

In the following exercises, determine if the given number is prime or composite.

251. 43 252. 67 253. 39

254. 53 255. 71 256. 119

257. 481 258. 221 259. 209

260. 359 261. 667 262. 1771

263.Banking Frank’s grandmother gave him $100 at his high school graduation Instead of spending it, Frank opened a bank account Every week, he added $15to the account The table shows how much money Frank had put in the account by the end of each week Complete the table by filling in the blanks

Weeks after graduation

Total number of dollars Frank

put in the account

Simplified Total

0 100 100

1 100 + 15 115

2 100 + 15 ⋅ 2 130

3 100 + 15 ⋅ 3

4 100 + 15 ⋅ [ ]

5 100 + [ ]

6 20

x

264.BankingIn March, Gina opened a Christmas club savings account at her bank She deposited $75 to open the account Every week, she added $20 to the account The table shows how much money Gina had put in the account by the end of each week Complete the table by filling in the blanks

Weeks after opening

the account

Total number of dollars Gina put in the account

Simplified Total

0 75 75

1 75 + 20 95

2 75 + 20 ⋅ 2 115

3 75 + 20 ⋅ 3

4 75 + 20 ⋅ [ ]

5 75 + [ ]

6 20

x

Writing Exercises

265.If a number is divisible by 2 and by 3, why is it also divisible by 6?

266.What is the difference between prime numbers and composite numbers?

Self Check

ⓐAfter completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

2.5 Prime Factorization and the Least Common Multiple Learning Objectives

By the end of this section, you will be able to:

Find the prime factorization of a composite number Find the least common multiple (LCM) of two numbers

Be Prepared!

Before you get started, take this readiness quiz Is 810 divisible by 2, 3, 5, 6, or 10?

If you missed this problem, reviewExample 2.44 Is 127 prime or composite?

If you missed this problem, reviewExample 2.47 Write 2 ⋅ ⋅ ⋅ 2 in exponential notation

If you missed this problem, reviewExample 2.5

Find the Prime Factorization of a Composite Number

In the previous section, we found the factors of a number Prime numbers have only two factors, the number 1 and the prime number itself Composite numbers have more than two factors, and every composite number can be written as a unique product of primes This is called theprime factorizationof a number When we write the prime factorization of a number, we are rewriting the number as a product of primes Finding the prime factorization of a composite number will help you later in this course

Prime Factorization

The prime factorization of a number is the product of prime numbers that equals the number

MANIPULATIVE MATHEMATICS

Doing the Manipulative Mathematics activity “Prime Numbers” will help you develop a better sense of prime numbers

You may want to refer to the following list of prime numbers less than 50 as you work through this section 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47

Prime Factorization Using the Factor Tree Method

One way to find the prime factorization of a number is to make a factor tree We start by writing the number, and then writing it as the product of two factors We write the factors below the number and connect them to the number with a small line segment—a “branch” of the factor tree

If a factor is prime, we circle it (like a bud on a tree), and not factor that “branch” any further If a factor is not prime, we repeat this process, writing it as the product of two factors and adding new branches to the tree

We continue until all the branches end with a prime When the factor tree is complete, the circled primes give us the prime factorization

For example, let’s find the prime factorization of 36. We can start with any factor pair such as 3 and 12. We write 3 and 12 below 36 with branches connecting them

The factor 3is prime, so we circle it The factor 4 is composite, and it factors into2 · 2. We write these factors under the 4. Since 2 is prime, we circle both 2s.

The prime factorization is the product of the circled primes We generally write the prime factorization in order from least to greatest

2 ⋅ ⋅ ⋅ 3

In cases like this, where some of the prime factors are repeated, we can write prime factorization in exponential form 2 ⋅ ⋅ ⋅ 3

22⋅ 32

Note that we could have started our factor tree with any factor pair of 36. We chose 12 and 3, but the same result would have been the same if we had started with 2 and 18, 4 and 9, or and 6.

EXAMPLE 2.48

Find the prime factorization of 48 using the factor tree method

HOW TO : :FIND THE PRIME FACTORIZATION OF A COMPOSITE NUMBER USING THE TREE METHOD Find any factor pair of the given number, and use these numbers to create two branches If a factor is prime, that branch is complete Circle the prime

If a factor is not prime, write it as the product of a factor pair and continue the process Write the composite number as the product of all the circled primes

Solution

We can start our tree using any factor pair of 48 Let's use and 24 We circle the because it is prime and so that branch is complete

Now we will factor 24 Let's use and

Neither factor is prime, so we not circle either We factor the 4, using and

We factor 6, using and

We circle the 2s and the since they are prime Now all of the branches end in a prime

Write the product of the circled numbers 2 ⋅ ⋅ ⋅ ⋅ 3 Write in exponential form 24⋅ 3

Check this on your own by multiplying all the factors together The result should be 48. TRY IT : :2.95 Find the prime factorization using the factor tree method: 80

TRY IT : :2.96 Find the prime factorization using the factor tree method: 60

EXAMPLE 2.49

We start with the factor pair and 21

Neither factor is prime so we factor them further

Now the factors are all prime, so we circle them

Then we write 84 as the product of all circled primes 2 ⋅ ⋅ ⋅ 7

22⋅ ⋅ 7 Draw a factor tree of 84.

TRY IT : :2.97 Find the prime factorization using the factor tree method: 126 TRY IT : :2.98 Find the prime factorization using the factor tree method: 294

Prime Factorization Using the Ladder Method

The ladder method is another way to find the prime factors of a composite number It leads to the same result as the factor tree method Some people prefer the ladder method to the factor tree method, and vice versa

To begin building the “ladder,” divide the given number by its smallest prime factor For example, to start the ladder for 36, we divide 36 by 2, the smallest prime factor of 36.

To add a “step” to the ladder, we continue dividing by the same prime until it no longer divides evenly

Then we divide by the next prime; so we divide 9 by 3.

We continue dividing up the ladder in this way until the quotient is prime Since the quotient, 3, is prime, we stop here Do you see why the ladder method is sometimes called stacked division?

The prime factorization is the product of all the primes on the sides and top of the ladder 2 ⋅ ⋅ ⋅ 3

22⋅ 32

EXAMPLE 2.50

Find the prime factorization of 120 using the ladder method

Solution

Divide the number by the smallest prime, which is

Continue dividing by until it no longer divides evenly

Divide by the next prime,

The quotient, 5, is prime, so the ladder is complete Write the prime factorization of 120 2 ⋅ ⋅ ⋅ ⋅ 5

23⋅ ⋅ 5 Check this yourself by multiplying the factors The result should be 120.

TRY IT : :2.99 Find the prime factorization using the ladder method: 80

TRY IT : :2.100 Find the prime factorization using the ladder method: 60 EXAMPLE 2.51

Find the prime factorization of 48 using the ladder method

HOW TO : :FIND THE PRIME FACTORIZATION OF A COMPOSITE NUMBER USING THE LADDER METHOD Divide the number by the smallest prime

Continue dividing by that prime until it no longer divides evenly Divide by the next prime until it no longer divides evenly Continue until the quotient is a prime

Write the composite number as the product of all the primes on the sides and top of the ladder

Divide the number by the smallest prime,

Continue dividing by until it no longer divides evenly

The quotient, 3, is prime, so the ladder is complete Write the prime factorization of 48 2 ⋅ ⋅ ⋅ ⋅ 3

24⋅ 3

TRY IT : :2.101 Find the prime factorization using the ladder method. 126

TRY IT : :2.102 Find the prime factorization using the ladder method 294

Find the Least Common Multiple (LCM) of Two Numbers

One of the reasons we look at multiples and primes is to use these techniques to find the least common multiple of two numbers This will be useful when we add and subtract fractions with different denominators

Listing Multiples Method

A common multiple of two numbers is a number that is a multiple of both numbers Suppose we want to find common multiples of 10 and 25. We can list the first several multiples of each number Then we look for multiples that are common to both lists—these are the common multiples

10:10, 20, 30, 40, 50, 60, 70, 80, 90, 100, 110, … 25:25, 50, 75, 100, 125, …

We see that 50 and 100 appear in both lists They are common multiples of 10 and 25. We would find more common multiples if we continued the list of multiples for each

The smallest number that is a multiple of two numbers is called theleast common multiple(LCM) So the least LCM of 10 and 25 is 50.

EXAMPLE 2.52

Find the LCM of 15 and 20 by listing multiples

Solution

List the first several multiples of 15 and of 20. Identify the first common multiple 15: 15, 30, 45, 60, 75, 90, 105, 120

20: 20, 40, 60, 80, 100, 120, 140, 160

HOW TO : :FIND THE LEAST COMMON MULTIPLE (LCM) OF TWO NUMBERS BY LISTING MULTIPLES List the first several multiples of each number

Look for multiples common to both lists If there are no common multiples in the lists, write out additional multiples for each number

Look for the smallest number that is common to both lists This number is the LCM

The smallest number to appear on both lists is 60, so 60 is the least common multiple of 15 and 20. Notice that 120 is on both lists, too It is a common multiple, but it is not the least common multiple

TRY IT : :2.103 Find the least common multiple (LCM) of the given numbers: 9 and 12

TRY IT : :2.104 Find the least common multiple (LCM) of the given numbers: 18 and 24

Prime Factors Method

Another way to find the least common multiple of two numbers is to use their prime factors We’ll use this method to find the LCM of 12 and 18.

We start by finding the prime factorization of each number

12 = ⋅ ⋅ 3 18 = ⋅ ⋅ 3

Then we write each number as a product of primes, matching primes vertically when possible 12 = ⋅ ⋅ 3

18 = ⋅ ⋅ 3

Now we bring down the primes in each column The LCM is the product of these factors

Notice that the prime factors of 12 and the prime factors of 18 are included in the LCM By matching up the common primes, each common prime factor is used only once This ensures that 36 is the least common multiple

EXAMPLE 2.53

Find the LCM of 15 and 18 using the prime factors method

Solution

Write each number as a product of primes

Write each number as a product of primes, matching primes vertically when possible

Bring down the primes in each column

Multiply the factors to get the LCM LCM = ⋅ ⋅ ⋅ 5 The LCM of 15 and 18 is 90 HOW TO : :FIND THE LCM USING THE PRIME FACTORS METHOD

Find the prime factorization of each number

Write each number as a product of primes, matching primes vertically when possible Bring down the primes in each column

Multiply the factors to get the LCM Step

TRY IT : :2.106 Find the LCM using the prime factors method. 15 and 35

EXAMPLE 2.54

Find the LCM of 50 and 100using the prime factors method

Solution

Write the prime factorization of each number

Write each number as a product of primes, matching primes vertically when possible

Bring down the primes in each column

Multiply the factors to get the LCM LCM = ⋅ ⋅ ⋅ 5

The LCM of 50 and 100 is 100

TRY IT : :2.107 Find the LCM using the prime factors method: 55, 88

TRY IT : :2.108 Find the LCM using the prime factors method: 60, 72

MEDIA : :ACCESS ADDITIONAL ONLINE RESOURCES

• Ex 1: Prime Factorization (http://openstaxcollege.org/l/24PrimeFactor1)

• Ex 2: Prime Factorization (http://openstaxcollege.org/l/24PrimeFactor2)

• Ex 3: Prime Factorization (http://openstaxcollege.org/l/24PrimeFactor3)

• Ex 1: Prime Factorization Using Stacked Division (http://openstaxcollege.org/l/24stackeddivis)

• Ex 2: Prime Factorization Using Stacked Division (http://openstaxcollege.org/l/24stackeddivis2)

• The Least Common Multiple (http://openstaxcollege.org/l/24LCM)

• Example: Determining the Least Common Multiple Using a List of Multiples (http://openstaxcollege.org/l/24LCM2)

Practice Makes Perfect

Find the Prime Factorization of a Composite Number

In the following exercises, find the prime factorization of each number using the factor tree method.

267. 86 268.78 269. 132

270. 455 271. 693 272. 420

273. 115 274.225 275. 2475

276.1560

In the following exercises, find the prime factorization of each number using the ladder method.

277. 56 278. 72 279. 168

280. 252 281. 391 282. 400

283. 432 284.627 285. 2160

286. 2520

In the following exercises, find the prime factorization of each number using any method.

287. 150 288. 180 289. 525

290. 444 291. 36 292. 50

293. 350 294. 144

Find the Least Common Multiple (LCM) of Two Numbers

In the following exercises, find the least common multiple (LCM) by listing multiples.

295. 8, 12 296.4, 3 297. 6, 15

298. 12, 16 299.30, 40 300. 20, 30

301. 60, 75 302.44, 55

In the following exercises, find the least common multiple (LCM) by using the prime factors method.

303. 8, 12 304. 12, 16 305. 24, 30

306. 28, 40 307.70, 84 308. 84, 90

In the following exercises, find the least common multiple (LCM) using any method.

309. 6, 21 310. 9, 15 311. 24, 30

312. 32, 40

313.Grocery shoppingHot dogs are sold in packages of ten, but hot dog buns come in packs of eight What is the smallest number of hot dogs and buns that can be purchased if you want to have the same number of hot dogs and buns? (Hint: it is the LCM!)

314. Grocery shopping Paper plates are sold in packages of 12 and party cups come in packs of 8. What is the smallest number of plates and cups you can purchase if you want to have the same number of each? (Hint: it is the LCM!)

Writing Exercises

315.Do you prefer to find the prime factorization of a composite number by using the factor tree method or the ladder method? Why?

316.Do you prefer to find the LCM by listing multiples or by using the prime factors method? Why?

Self Check

ⓐAfter completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

coefficient composite number divisibility equation evaluate expressions

least common multiple like terms

multiple of a number prime factorization prime number

solution of an equation term

CHAPTER REVIEW

KEY TERMS

The constant that multiplies the variable(s) in a term is called the coefficient A composite number is a counting number that is not prime If a number m is a multiple of n, then we say that m is divisible by n An equation is made up of two expressions connected by an equal sign

To evaluate an algebraic expression means to find the value of the expression when the variable is replaced by a given number

An expression is a number, a variable, or a combination of numbers and variables and operation symbols The smallest number that is a multiple of two numbers is called the least common multiple (LCM)

Terms that are either constants or have the same variables with the same exponents are like terms A number is a multiple of n if it is the product of a counting number and n

The prime factorization of a number is the product of prime numbers that equals the number A prime number is a counting number greater than whose only factors are and itself

A solution to an equation is a value of a variable that makes a true statement when substituted into the equation The process of finding the solution to an equation is called solving the equation

A term is a constant or the product of a constant and one or more variables

KEY CONCEPTS

2.1 Use the Language of Algebra

Operation Notation Say: The result is…

Addition a + b a plus b the sum of a and b Multiplication a · b, (a)(b), (a)b, a(b) a times b The product of a and b Subtraction a − b a minus b the difference of a and b Division a ÷ b, a / b, ab, b a a divided by b The quotient of a and b

• Equality Symbol

◦ a = b is read as a is equal to b

◦ The symbol = is called the equal sign • Inequality

◦ a < b is read a is less than b

Algebraic Notation Say

a = b a is equal to b a ≠ b a is not equal to b a < b a is less than b a > b a is greater than b a ≤ b a is less than or equal to b a ≥ b a is greater than or equal to b

Table 2.77

• Exponential Notation

◦ For any expression an is a factor multiplied by itself n times, if n is a positive integer ◦ an means multiply n factors of a

◦ The expression of an is read a to the nth power

Order of OperationsWhen simplifying mathematical expressions perform the operations in the following order: • Parentheses and other Grouping Symbols: Simplify all expressions inside the parentheses or other grouping

symbols, working on the innermost parentheses first • Exponents: Simplify all expressions with exponents

• Multiplication and Division: Perform all multiplication and division in order from left to right These operations have equal priority

• Addition and Subtraction: Perform all addition and subtraction in order from left to right These operations have equal priority

2.2 Evaluate, Simplify, and Translate Expressions

• Combine like terms Identify like terms

Rearrange the expression so like terms are together Add the coefficients of the like terms

2.3 Solving Equations Using the Subtraction and Addition Properties of Equality

• Determine whether a number is a solution to an equation Substitute the number for the variable in the equation Simplify the expressions on both sides of the equation

Determine whether the resulting equation is true If it is true, the number is a solution If it is not true, the number is not a solution

• Subtraction Property of Equality ◦ For any numbers a, b, and c, Step

Step Step

if a = b then a − b = b − c

• Solve an equation using the Subtraction Property of Equality

Use the Subtraction Property of Equality to isolate the variable Simplify the expressions on both sides of the equation Check the solution

• Addition Property of Equality

◦ For any numbers a, b, and c,

if a = b then a + b = b + c

• Solve an equation using the Addition Property of Equality

Use the Addition Property of Equality to isolate the variable Simplify the expressions on both sides of the equation Check the solution

2.4 Find Multiples and Factors

Divisibility Tests A number is divisible by

2 if the last digit is0, 2, 4, 6,or8 3 if the sum of the digits is divisible by3 5 if the last digit is5or0

6 if divisible by both2and3 10 if the last digit is0

• Factors If a ⋅ b = m, then a and bare factors of m, and m is the product of a and b • Find all the factors of a counting number

Divide the number by each of the counting numbers, in order, until the quotient is smaller than the divisor a If the quotient is a counting number, the divisor and quotient are a pair of factors

b If the quotient is not a counting number, the divisor is not a factor List all the factor pairs

Write all the factors in order from smallest to largest • Determine if a number is prime

Test each of the primes, in order, to see if it is a factor of the number

Start with and stop when the quotient is smaller than the divisor or when a prime factor is found If the number has a prime factor, then it is a composite number If it has no prime factors, then the number is prime

• Find the prime factorization of a composite number using the tree method

Find any factor pair of the given number, and use these numbers to create two branches If a factor is prime, that branch is complete Circle the prime

If a factor is not prime, write it as the product of a factor pair and continue the process Write the composite number as the product of all the circled primes

• Find the prime factorization of a composite number using the ladder method Divide the number by the smallest prime

Continue dividing by that prime until it no longer divides evenly Divide by the next prime until it no longer divides evenly Continue until the quotient is a prime

Write the composite number as the product of all the primes on the sides and top of the ladder • Find the LCM using the prime factors method

Find the prime factorization of each number

Write each number as a product of primes, matching primes vertically when possible Bring down the primes in each column

Multiply the factors to get the LCM • Find the LCM using the prime factors method

Find the prime factorization of each number

Write each number as a product of primes, matching primes vertically when possible Bring down the primes in each column

Multiply the factors to get the LCM

REVIEW EXERCISES

2.1 Use the Language of Algebra

Use Variables and Algebraic Symbols

In the following exercises, translate from algebra to English.

317. 3 ⋅ 8 318. 12 − x 319. 24 ÷ 6

320. 9 + 2a 321. 50 ≥ 47 322. 3y < 15

323. n + = 13 324. 32 − k = 7

Identify Expressions and Equations

In the following exercises, determine if each is an expression or equation.

325. 5 + u = 84 326. 36 − 6s 327. 4y − 11

328. 10x = 120

Simplify Expressions with Exponents

In the following exercises, write in exponential form.

329. 2 ⋅ ⋅ 2 330. a ⋅ a ⋅ a ⋅ a ⋅ a 331. x ⋅ x ⋅ x ⋅ x ⋅ x ⋅ x 332. 10 ⋅ 10 ⋅ 10

In the following exercises, write in expanded form.

333. 84 334. 36 335. y5

336. n4

In the following exercises, simplify each expression.

337. 34 338. 106 339. 27

340. 43

Simplify Expressions Using the Order of Operations

In the following exercises, simplify.

341. 10 + ⋅ 5 342. (10 + 2) ⋅ 5 343. (30 + 6) ÷ 2

344. 30 + ÷ 2 345. 72+ 52 346. (7 + 5)2

347. 4 + 3(10 − 1) 348. (4 + 3)(10 − 1)

2.2 Evaluate, Simplify, and Translate Expressions

Evaluate an Expression

In the following exercises, evaluate the following expressions.

349. 9x − when x = 7 350. y3when y = 5 351. 3a − 4b when

a = 10, b = 1

352. bhwhen b = 7, h = 8

Identify Terms, Coefficients and Like Terms

In the following exercises, identify the terms in each expression.

353. 12n2+ 3n + 1 354. 4x3+ 11x + 3

In the following exercises, identify the coefficient of each term.

355. 6y 356. 13x2

In the following exercises, identify the like terms.

357. 5x2, 3, 5y2, 3x, x, 4 358. 8, 8r2, 8r, 3r, r2, 3s

Simplify Expressions by Combining Like Terms

In the following exercises, simplify the following expressions by combining like terms.

359. 15a + 9a 360. 12y + 3y + y 361. 4x + 7x + 3x

362. 6 + 5c + 3 363. 8n + + 4n + 9 364. 19p + + 4p − + 3p

In the following exercises, translate the following phrases into algebraic expressions.

367. the difference of x and 6 368. the sum of 10 and twice a 369. the product of 3n and 9 370. the quotient of s and 4 371. 5 times the sum of y and 1 372. 10 less than the product of

5 and z

373. Jack bought a sandwich and a coffee The cost of the sandwich was $3 more than the cost of the coffee Call the cost of the coffee

c. Write an expression for the cost of the sandwich

374. The number of poetry books on Brianna’s bookshelf is 5 less than twice the number of novels Call the number of novels n. Write an expression for the number of poetry books

2.3 Solve Equations Using the Subtraction and Addition Properties of Equality

Determine Whether a Number is a Solution of an Equation

In the following exercises, determine whether each number is a solution to the equation.

375. y + 16 = 40

ⓐ24 ⓑ56

376. d − = 21

ⓐ15 ⓑ27

377. 4n + 12 = 36

ⓐ6 ⓑ12 378. 20q − 10 = 70

ⓐ3 ⓑ4

379. 15x − = 10x + 45

ⓐ2 ⓑ10

380. 22p − = 18p + 86

ⓐ4 ⓑ23

Model the Subtraction Property of Equality

In the following exercises, write the equation modeled by the envelopes and counters and then solve the equation using the subtraction property of equality.

381. 382.

Solve Equations using the Subtraction Property of Equality

In the following exercises, solve each equation using the subtraction property of equality.

383. c + = 14 384. v + = 150 385. 23 = x + 12 386. 376 = n + 265

Solve Equations using the Addition Property of Equality

In the following exercises, solve each equation using the addition property of equality.

Translate English Sentences to Algebraic Equations

In the following exercises, translate each English sentence into an algebraic equation.

391. The sum of 7 and 33 is equal to 40.

392. The difference of 15 and 3 is equal to 12.

393. The product of 4 and 8 is equal to 32.

394. The quotient of 63 and 9 is equal to 7.

395. Twice the difference of n and 3 gives 76.

396. The sum of five times y and 4 is 89.

Translate to an Equation and Solve

In the following exercises, translate each English sentence into an algebraic equation and then solve it.

397. Eight more than xis equal to

35. 398. 21 less than a is 11. 399.is 57.The difference of q and 18 400. The sum of m and 125 is

240.

Mixed Practice

In the following exercises, solve each equation.

401. h − 15 = 27 402. k − 11 = 34 403. z + 52 = 85 404. x + 93 = 114 405. 27 = q + 19 406. 38 = p + 19

407. 31 = v − 25 408. 38 = u − 16

2.4 Find Multiples and Factors

Identify Multiples of Numbers

In the following exercises, list all the multiples less than 50 for each of the following.

409. 3 410. 2 411. 8

412. 10

Use Common Divisibility Tests

In the following exercises, using the divisibility tests, determine whether each number is divisible by 2, by 3, by 5, by 6, and by 10.

413. 96 414. 250 415. 420

416. 625

Find All the Factors of a Number

In the following exercises, find all the factors of each number.

417. 30 418. 70 419. 180

420. 378

Identify Prime and Composite Numbers

In the following exercises, identify each number as prime or composite.

2.5 Prime Factorization and the Least Common Multiple

Find the Prime Factorization of a Composite Number

In the following exercises, find the prime factorization of each number.

425. 84 426. 165 427. 350

428. 572

Find the Least Common Multiple of Two Numbers

In the following exercises, find the least common multiple of each pair of numbers.

429. 9, 15 430. 12, 20 431. 25, 35

432. 18, 40

Everyday Math

PRACTICE TEST

In the following exercises, translate from an algebraic equation to English phrases.

434. 6 ⋅ 4 435. 15 − x

In the following exercises, identify each as an expression or equation.

436. 5 ⋅ + 10 437. x + = 9 438. 3 ⋅ 11 = 33

439.

ⓐ Write

n ⋅ n ⋅ n ⋅ n ⋅ n ⋅ n in exponential form

ⓑ Write 35 in

expanded form and then simplify

In the following exercises, simplify, using the order of operations.

440. 4 + ⋅ 5 441. (8 + 1) ⋅ 4 442. 1 + 6(3 − 1)

443. (8 + 4) ÷ + 1 444. (1 + 4)2 445. 5[2 + 7(9 − 8)]

In the following exercises, evaluate each expression.

446. 8x − when x = 4 447. y3when y = 5 448. 6a − 2b when a = 5, b = 7 449. hw when h = 12, w = 3 450. Simplify by combining like

terms

ⓐ6x + 8x

ⓑ9m + 10 + m + 3

In the following exercises, translate each phrase into an algebraic expression.

451. 5 more than x 452. the quotient of 12 and y 453. three times the difference of

a and b

454. Caroline has 3 fewer earrings on her left ear than on her right ear Call the number of earrings on her right ear, r. Write an expression for the number of earrings on her left ear

In the following exercises, solve each equation.

455. n − = 25 456. x + 58 = 71

In the following exercises, translate each English sentence into an algebraic equation and then solve it.

457. 15 less than y is 32. 458. the sum of a and 129 is 164.

459. List all the multiples of 4, that are less than 50.

460. Find all the factors of 90. 461. Find the prime factorization

Figure 3.1 The peak of Mount Everest (credit: Gunther Hagleitner, Flickr)

Chapter Outline

3.1Introduction to Integers

3.2Add Integers

3.3Subtract Integers

3.4Multiply and Divide Integers

3.5Solve Equations Using Integers; The Division Property of Equality Introduction

At over 29,000 feet, Mount Everest stands as the tallest peak on land Located along the border of Nepal and China, Mount Everest is also known for its extreme climate Near the summit, temperatures never rise above freezing Every year, climbers from around the world brave the extreme conditions in an effort to scale the tremendous height Only some are successful Describing the drastic change in elevation the climbers experience and the change in temperatures requires using numbers that extend both above and below zero In this chapter, we will describe these kinds of numbers and operations using them

3.1 Introduction to Integers Learning Objectives

By the end of this section, you will be able to:

Locate positive and negative numbers on the number line Order positive and negative numbers

Find opposites

Simplify expressions with absolute value

Translate word phrases to expressions with integers

Be Prepared!

Before you get started, take this readiness quiz Plot 0, 1, and 3 on a number line

If you missed this problem, reviewExample 1.1 Fill in the appropriate symbol: (=, <, or >): _4

If you missed this problem, reviewExample 2.3

Locate Positive and Negative Numbers on the Number Line

Do you live in a place that has very cold winters? Have you ever experienced a temperature below zero? If so, you are INTEGERS

measured in degrees below zero and can be described by negative numbers For example, −1°F (read as “negative one degree Fahrenheit”) is 1 degree below 0. A minus sign is shown before a number to indicate that it is negative.Figure 3.2shows −20°F, which is 20 degrees below 0.

Figure 3.2 Temperatures below zero are described by negative numbers

Temperatures are not the only negative numbers A bank overdraft is another example of a negative number If a person writes a check for more than he has in his account, his balance will be negative

Elevations can also be represented by negative numbers The elevation at sea level is 0 feet. Elevations above sea level are positive and elevations below sea level are negative The elevation of the Dead Sea, which borders Israel and Jordan, is about 1,302 feet below sea level, so the elevation of the Dead Sea can be represented as −1,302 feet. SeeFigure 3.3

Figure 3.3 The surface of the Mediterranean Sea has an elevation of 0 ft. The diagram shows that nearby mountains have higher (positive) elevations whereas the Dead Sea has a lower (negative) elevation

Depths below the ocean surface are also described by negative numbers A submarine, for example, might descend to a depth of 500 feet. Its position would then be −500 feet as labeled inFigure 3.4

Figure 3.4 Depths below sea level are described by negative numbers A submarine 500 ft below sea level is at −500 ft.

number is assumed to be positive

Figure 3.5

Now we need to extend the number line to include negative numbers We mark several units to the left of zero, keeping the intervals the same width as those on the positive side We label the marks with negative numbers, starting with −1 at the first mark to the left of 0, −2 at the next mark, and so on SeeFigure 3.6

Figure 3.6 On a number line, positive numbers are to the right of zero Negative numbers are to the left of zero What about zero? Zero is neither positive nor negative

The arrows at either end of the line indicate that the number line extends forever in each direction There is no greatest positive number and there is no smallest negative number

MANIPULATIVE MATHEMATICS

Doing the Manipulative Mathematics activity "Number Line-part 2" will help you develop a better understanding of integers

EXAMPLE 3.1

Plot the numbers on a number line:

ⓐ 3 ⓑ−3 ⓒ−2

Solution

Draw a number line Mark 0in the center and label several units to the left and right

ⓐ To plot 3, start at 0 and count three units to the right Place a point as shown inFigure 3.7

Figure 3.7

ⓑ To plot −3, start at 0 and count three units to the left Place a point as shown inFigure 3.8

Figure 3.8

ⓒ To plot −2, start at 0 and count two units to the left Place a point as shown inFigure 3.9

ⓐ 1 ⓑ −1 ⓒ −4 TRY IT : :3.2 Plot the numbers on a number line.

ⓐ −4 ⓑ 4 ⓐ −1

Order Positive and Negative Numbers

We can use the number line to compare and order positive and negative numbers Going from left to right, numbers increase in value Going from right to left, numbers decrease in value SeeFigure 3.10

Figure 3.10

Just as we did with positive numbers, we can use inequality symbols to show the ordering of positive and negative numbers Remember that we use the notation a < b (read a is less than b) when a is to the left of b on the number line We write a > b (read a is greater than b) when a is to the right of b on the number line This is shown for the numbers 3and 5 inFigure 3.11

Figure 3.11 The number 3is to the left of

5on the number line So 3is less than

5, and 5 is greater than3.

The numbers lines to follow show a few more examples

ⓐ

4 is to the right of 1on the number line, so 4 > 1. 1 is to the left of 4on the number line, so 1 < 4.

ⓑ

−2 is to the left of 1 on the number line, so −2 < 1. 1 is to the right of −2 on the number line, so 1 > −2.

ⓒ

EXAMPLE 3.2

Order each of the following pairs of numbers using < or >:

ⓐ 14 _6 ⓑ −1 _9 ⓒ −1 _−4 ⓓ2 _−20

Solution

Begin by plotting the numbers on a number line as shown inFigure 3.12

Figure 3.12

ⓐCompare 14 and 14 _6 14 is to the right of on the number line 14 > 6

ⓑCompare −1 and −1 _9 −1 is to the left of on the number line −1 < 9

ⓒCompare −1 and −4 −1 _−4 −1 is to the right of −4 on the number line −1 > −4

ⓓCompare and −20 −2 _−20 is to the right of −20 on the number line 2 > −20

TRY IT : :3.3 Order each of the following pairs of numbers using < or >.

ⓐ 15 _7 ⓑ −2 _5 ⓒ −3 _−7 ⓓ 5 _−17

TRY IT : :3.4 Order each of the following pairs of numbers using < or>.

ⓐ 8 _13 ⓑ3 _−4 ⓒ−5 _−2 ⓓ 9 _−21

Find Opposites

Figure 3.13 Opposite

The opposite of a number is the number that is the same distance from zero on the number line, but on the opposite side of zero

EXAMPLE 3.3

Find the opposite of each number:

ⓐ 7 ⓑ−10

Solution

ⓐThe number −7 is the same distance from 0 as 7, but on the opposite side of 0. So −7 is the opposite of 7 as shown inFigure 3.14

Figure 3.14

ⓑThe number 10 is the same distance from 0as −10, but on the opposite side of 0. So 10 is the opposite of −10 as shown inFigure 3.15

Figure 3.15

TRY IT : :3.5 Find the opposite of each number:

ⓐ 4 ⓑ−3

TRY IT : :3.6 Find the opposite of each number:

ⓐ 8 ⓑ −5

The specific meaning becomes clear by looking at how it is used You have seen the symbol “−”, in three different ways 10 − 4 Between two numbers, the symbol indicates the operation of subtraction.We read 10 − 4 as 10minus 4.

−8 In front of a number, the symbol indicates a negative number.We read −8 asnegative eight. −x In front of a variable or a number, it indicates the opposite.We read−x asthe opposite of x.

− (−2) Here we have two signs The sign in the parentheses indicates that the number is negative 2.The sign outside the parentheses indicates the opposite We read − (−2) asthe opposite of −2.

Opposite Notation

−a means the opposite of the number a

The notation −ais readthe opposite of a.

EXAMPLE 3.4 Simplify: −(−6).

Solution

−(−6)

The opposite of −6 is 6. 6

TRY IT : :3.7 Simplify:

−(−1) TRY IT : :3.8 Simplify:

−(−5)

Integers

The set of counting numbers, their opposites, and 0 is the set of integers

Integers

Integersare counting numbers, their opposites, and zero

… −3, −2, −1, 0, 1, 2, 3… We must be very careful with the signs when evaluating the opposite of a variable

EXAMPLE 3.5 Evaluate −x :

ⓐTo evaluate −x when x = 8, substitute 8 for x

−x

Simplify −8

ⓑTo evaluate −x when x = −8, substitute −8 for x

−x

Simplify 8

TRY IT : :3.9 Evaluate −n :

ⓐ when n = 4 ⓑ when n = −4 TRY IT : :3.10 Evaluate: −m :

ⓐ when m = 11 ⓑ when m = −11

Simplify Expressions with Absolute Value

We saw that numbers such as 5and −5 are opposites because they are the same distance from 0 on the number line They are both five units from 0. The distance between 0 and any number on the number line is called theabsolute valueof that number Because distance is never negative, the absolute value of any number is never negative

The symbol for absolute value is two vertical lines on either side of a number So the absolute value of 5 is written as

|5|, and the absolute value of −5 is written as |−5| as shown inFigure 3.16

Figure 3.16 Absolute Value

The absolute value of a number is its distance from 0 on the number line The absolute value of a number nis written as |n|.

|n| ≥ for all numbers EXAMPLE 3.6

ⓐ |3| ⓑ |−44| ⓒ |0|

Solution

ⓐ

|3|

3 is units from zero 3

ⓑ

|−44|

−44 is 44 units from zero 44

ⓒ

|0|

0 is already at zero 0

TRY IT : :3.11 Simplify:

ⓐ |12| ⓑ − |−28|

TRY IT : :3.12 Simplify:

ⓐ |9| ⓑ (b) −|37|

We treat absolute value bars just like we treat parentheses in the order of operations We simplify the expression inside first

EXAMPLE 3.7 Evaluate:

ⓐ |x|when x = −35 ⓑ|−y|when y = −20 ⓒ −|u|when u = 12 ⓓ −|p|when p = −14

Solution

ⓐTo find |x| when x = −35 :

|x|

ⓑTo find | − y| when y = −20 :

| − y|

Simplify |20|

Take the absolute value 20

ⓒTo find − |u| when u = 12 :

− |u|

Take the absolute value −12

ⓓTo find − |p| when p = −14 :

− |p|

Take the absolute value −14

Notice that the result is negative only when there is a negative sign outside the absolute value symbol TRY IT : :3.13

Evaluate:

ⓐ|x|when x = −17 ⓑ |−y|when y = −39 ⓒ −|m|when m = 22

ⓓ −|p|when p = −11 TRY IT : :3.14

ⓐ|y|when y = −23 ⓑ |−y|when y = −21 ⓒ −|n|when n = 37

ⓓ −|q|when q = −49 EXAMPLE 3.8

Fill in <, >, or = for each of the following:

ⓐ |−5| _ −|−5| ⓑ 8 _ −|−8| ⓒ −9 _ −|−9| ⓓ −|−7| _−7

Solution